18.8: Checklists for Parenteral Medication Administration
Checklist for Parenteral Site Identification
Use the checklist below to review the steps for completion of “Parenteral Site Identification.”
Directions: Identify parenteral injection sites, needle size/gauge, injection angle, and the appropriate amount that can be administered in each of the parenteral routes: intradermal, subcutaneous, and intramuscular.
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Describe the appropriate needle gauge, length, number of cc’s, and angle for an intradermal injection:
- 25-27G
- 3/8″ to 5/8″
- 0.1 mL (for TB testing)
- 5 to 15 degree angle
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Demonstrate locating the intradermal injection sites on a peer:
- Upper third of the forearm
- Outer aspects upper arms
- Between scapula
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Describe the appropriate needle gauge, length, number of mL, and angle for a subcutaneous injection:
- 25-31G
- ½” to 5/8″
- Up to 1 mL
- 45 to 90 degree angle
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Demonstrate locating the subcutaneous injection sites on a peer:
- Anterior thighs
- Abdomen
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Describe the appropriate needle gauge, length, number of mL, and angle for an adult intramuscular:
- 18-25G
- ½” – 1 ½” (based on age/size of patient and site used)
- <0.5 – 1 mL (infants and children), 2-5 mL (adults)
- 90-degree angle
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Demonstrate locating the intramuscular injection sites on a peer:
- Ventrogluteal
- Vastus laterus
- Deltoid
- Explain how you would modify assessment techniques to reflect variations across the life span and body size variations.
Checklist for Parenteral Medication Injections
Use the checklist below to review the steps for completion of “Parenteral Medication Injections.”
Steps
Disclaimer: Always review and follow agency policy regarding this specific skill.
Special Considerations:
- Plan medication administration to avoid disruption.
- Dispense medication in a quiet area.
- Avoid conversation with others.
- Follow agency’s no-interruption zone policy.
- Prepare medications for ONE patient at a time.
- Plan for disposal of sharps in an appropriate sharps disposal container.
- Check the orders and MAR for accuracy and completeness; clarify any unclear orders.
- Review pertinent information related to the medications: labs, last time medication was given, and medication information: generic name, brand name, dose, route, time, class, action, purpose, side effects, contraindications, and nursing considerations.
- Gather available supplies: correctly sized syringes and needles appropriate for medication, patient’s size, and site of injection; diluent (if required); tape or patient label for each syringe; nonsterile gloves; sharps container; and alcohol wipes.
- Perform hand hygiene.
- While withdrawing medication from the medication dispensing system, perform the first check of the six rights of medication administration. Check expiration date and perform any necessary calculations.
- Select the correct type of syringe and needle size appropriate for the medication, patient size, and site of injection.
Preparing the Medication for Administration
- Scrub the top of the vial of the correct medication. State the correct dose to be drawn.
- Remove the cap from the needle. Pull back on the plunger to draw air into the syringe equal to the dose.
- With the vial on a flat surface, insert the needle. Invert the vial and withdraw the correct amount of the medication. Expel any air bubbles. Remove the needle from the vial.
- Using the scoop method, recap the needle.
- Perform the second check of the six rights of medication administration, looking at the vial, syringe, and MAR.
- Label the syringe with the name of the drug and dose.
Additional Preparation Steps When Mixing Two Types of Insulin in One Syringe
Intermediate-Acting (NPH) and Short-Acting (Regular) insulins
- Place the vials side by side on a flat surface: NPH on left and regular insulin on the right.
- With an alcohol pad, scrub off the vial top of the NPH insulin. Using a new alcohol pad, scrub the vial top of the regular insulin. Discard any prep pads.
- Select the correct insulin syringe that will exactly measure the TOTAL dose of the amount of NPH and regular doses (30- and 50-unit syringes measure single units; 100-unit syringes only measure even numbered doses).
- Pull back on the plunger to draw air into the syringe equal to the dose of NPH insulin.
- With the NPH vial on a flat surface, remove the cap from the syringe, insert the needle into the NPH vial, and inject air. Do not let the tip of the needle touch the insulin solution. Withdraw the needle.
- Pull back on the plunger to draw air into the syringe equal to the dose of the regular insulin.
- With the regular vial on a flat surface, remove the cap from the syringe, insert the needle into the regular vial, and inject air.
- With the needle still in the vial, invert the regular insulin vial and withdraw the correct dose. Remove the needle from the vial. Cap the needle using the scoop method.
- Roll the NPH insulin vial between your hands to mix the solution. Uncap the needle and insert the needle into the NPH insulin vial. Withdraw the correct amount of NPH insulin. Withdraw the needle and recap using the scoop method.
- Perform the second medication check of the combined dose looking at the vial, syringe, and MAR, verifying all the rights.
- Label the syringe with the name of the combined medications and doses.
Alternative Preparation Using an Insulin Pen
- Select the correct insulin pen to be used for the injection. Identify the dose to be given.
- Remove the cap from the insulin pen and clean the top (hub) with an alcohol prep pad. Attach the insulin pen needle without contaminating the needle or pen hub.
- Turn the dial to two units and push the injection button to prime the pen.
- Turn the dial to the correct dose.
- Perform the second medication check looking at the insulin pen and MAR, verifying all the rights.
Administration of Parenteral Medication
- Knock, enter the room, greet the patient, and provide for privacy.
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Perform safety steps:
- Perform hand hygiene.
- Check the room for transmission-based precautions.
- Introduce yourself, your role, the purpose of your visit, and an estimate of the time it will take.
- Confirm patient ID using two patient identifiers (e.g., name and date of birth).
- Explain the process to the patient and ask if they have any questions.
- Be organized and systematic.
- Use appropriate listening and questioning skills.
- Listen and attend to patient cues.
- Ensure the patient’s privacy and dignity.
- Assess ABCs.
- Perform the third check of the six rights of medication administration at the patient’s bedside after performing patient identification.
- Perform the following steps according to the type of parenteral medication.
INTRADERMAL – Administration of a TB Test
- Correctly identify the sites and verbalize the landmarks used for intradermal injections.
- Select the correct site for the TB test, verbalizing the anatomical landmarks and skin considerations.
- Put on nonsterile gloves if contact with blood or body fluids is likely or if your skin or the patient’s skin isn’t intact.
- Use an alcohol swab in a circular motion to clean the skin at the site; place the pad above the site to mark the site, if desired.
- Using the nondominant hand, gently pull the skin away from the site.
- Insert the needle with the bevel facing upward, slowly at a 5- to 15-degree angle, and then advance no more than an eighth of an inch to cover the bevel.
- Use the thumb of the nondominant hand to push on the plunger to slowly inject the medication. Inspect the site, noting if a small bleb forms under the skin surface.
- Carefully withdraw the needle straight back out of the insertion site so not to disturb the bleb (do not massage or cover the site).
- Activate the safety feature of the needle and place the syringe in the sharps container.
- Teach the patient to return for a TB skin test reading in 48-72 hours and not to press on the site or apply a Band-Aid.
SUBCUTANEOUS – Administration of Insulin in a Syringe
- Correctly identify the sites and verbalize the landmarks used for subcutaneous injections. Ask the patient regarding a preferred site of medication administration.
- Put on nonsterile gloves if contact with blood or body fluids is likely or if your skin or the patient’s skin isn’t intact.
- Select an appropriate site and clean with an alcohol prep in a circular motion. Place the pad above the site to mark the location, if desired. Remove the cap from the needle without contaminating the needle.
- Pinch approximately an inch of subcutaneous tissue, creating a skinfold.
- Inject the needle at 90-degree angle, release the patient’s skin, and inject the medication. Withdraw the needle.
- Activate the safety feature of the needle and place the syringe in a sharps container.
SUBCUTANEOUS – Administration with an Insulin Pen
- Select the site and clean with an alcohol prep in a circular motion. Place the pad above the site to mark the location, if desired. Remove the cap from the needle without contaminating the needle.
- Put on nonsterile gloves if contact with blood or body fluids is likely or if your skin or the patient’s skin isn’t intact.
- Pinch approximately an inch of subcutaneous tissue, creating a skinfold.
- Inject the needle quickly at a 90-degree angle, continue to hold the skinfold, and inject the medication. After the medication is injected, count to 10, remove the needle, and release the skinfold.
- Dispose of the needle in a sharps container. Replace the top cap to the insulin pen.
INTRAMUSCULAR – Deltoid
- Correctly identify the site and verbalize the landmarks used for a deltoid injection.
- Put on nonsterile gloves if contact with blood or body fluids is likely or if your skin or the patient’s skin isn’t intact.
- Use an alcohol swab in a circular motion to clean the skin at the site. Place a pad above the site to mark the location. Remove the cap from the needle without contaminating the needle.
- Depending on the muscle mass of the deltoid, either grasp the body of the muscle between the thumb and forefingers of the nondominant hand or spread the skin taut.
- Inject the needle at a 90-degree angle.
- Follow agency policy and manufacturer recommendations regarding aspiration.
- Continue to hold the muscle fold and inject the medication. After the medication is injected, count to 10, remove the needle, and release the muscle fold.
- Activate the safety on the syringe. Place the syringe in a sharps container.
INTRAMUSCULAR – Vastus Lateralis
- Correctly identify the site and verbalize the landmarks to locate the vastus lateralis site.
- Put on nonsterile gloves if contact with blood or body fluids is likely or if your skin or the patient’s skin isn’t intact.
- Use an alcohol swab in a circular motion to clean the skin at the site. Place the pad above the site to mark the location. Remove the cap from the needle without contaminating the needle.
- Depending on the muscle mass of the vastus lateralis, either grasp the body of the muscle between the thumb and forefingers of the nondominant hand or spread the skin taut.
- Inject the needle at a 90-degree angle.
- Follow agency policy and manufacturer recommendations regarding aspiration.
- Continue to hold the muscle fold and inject the medication. After the medication is injected, count to 10, remove the needle, and release the muscle fold.
- Activate the safety on the syringe. Put the needle in a sharps container.
INTRAMUSCULAR – Ventrogluteal (Using the Z-track Technique)
- Correctly identify and verbalize the landmarks used to locate the ventrogluteal site.
- Put on nonsterile gloves if contact with blood or body fluids is likely or if your skin or the patient’s skin isn’t intact.
- Use an alcohol swab in a circular motion to clean the skin at the site and place a pad above the site to mark the location. Remove the cap from the needle without contaminating the needle.
- Place the ulnar surface of the hand approximately 1 – 3 inches from the selected site; press down and pull the skin and subcutaneous tissue to the side or downward.
- Maintaining tissue traction, hold the syringe like a dart and insert the needle into the skin at 90 degrees.
- Maintaining tissue traction, use the available thumb and index finger to help stabilize the syringe.
- Follow agency policy and manufacturer recommendations regarding aspiration. If aspiration is required, pull back the plunger and observe for blood return. If there is no blood return, inject the medication. If blood return is observed, remove the needle, and prepare a new medication.
- Maintaining tissue traction, wait 10 seconds with the needle still in the skin to allow the muscle to absorb the medication. Withdraw the needle from the site and then release traction. Do not rub/massage the site.
- Activate the safety feature of the needle; place in a sharps container.
Following Conclusion of All Injections
- Assess site; apply Band-Aid if necessary and appropriate.
- Remove gloves. Perform hand hygiene.
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Ensure safety measures before leaving the room:
- CALL LIGHT: Within reach
- BED: Low and locked (in lowest position and brakes on)
- SIDE RAILS: Secured
- TABLE: Within reach
- ROOM: Risk-free for falls (scan room and clear any obstacles)
- Document medication administered, including the site used for the injection.


8.8: Equal Employment Opportunity Law
- Explain the equal employment opportunity law
Retail managers and human resource management within a retail company are most likely familiar with or at some time become aware of the Equal Employment Opportunity Act (EEOA) as implemented by Congress. [1] This act is an amendment to the Civil Rights Act of 1964 which prohibits discrimination based on color, race, religion, national origin and gender. Congress further added to this act by passing the Age Discrimination in Employment Act of 1967 which also prohibits discrimination based on age and concentrating on the protection of individual employees ages 40 to 65 years of age. [2] Another act implemented by Congress was the Americans with Disabilities Act of 1990 which requires employers to make reasonable accommodations to disabled individuals.
A regulatory administrative agency governing the enforcement of the EEOA was established by the federal government. This agency is called the Equal Employment Opportunity Commission (EEOC). [3] The EEOC has the power to seek statutory remedies for individuals whose rights as set forth by the EEOA have been egregiously violated. These remedies can include but are not limited to punitive and compensatory damages and back-pay (depending on the case). However, the EEOC is not an answer or remedy necessarily for everyone who feels that they have became a victim of prejudice in the workplace as they typically only file suit in less than one percent of charges per year. [4]
If an employee feels that they have been discriminated against and moves forward with filing suit, the burden of proof relies on the plaintiff and he or she must build a “prima facie” case of discrimination. This means that the plaintiff must prove the employers discriminatory motive as prohibited by the EEOA. For example, written proof in the form of email or text explaining to the employee that they did not receive a promotion solely based off of the fact that they are a woman or a minority could be considered proof enough for the EEOC to pick up the suit. Typically the EEOC establishes proving discrimination based off of the treatment the employee received, the impact it had on them and/or if there was harassment involved. Harassment is another way to prove discrimination. In this instance it is a bit more complicated to build a “prima facie” case against a potential violator of this type of discrimination which will be addressed further in the section on sexual harassment laws.
For an employee to prove discrimination as defined by the EEOC the individual employee (or alleged victim in this instance) must be capable of establishing that “he or she was treated differently than other employees who were similarly situated, and that the difference was based on a protected characteristic.” [5] The EEOC has various types of reporting forms which are used and kept as record if a case is filed against a company for discrimination. Filing a charge against an employer would require an individual employee to sign a sworn statement naming the offender and detailing the discriminatory offense. Once the EEOC has received a complaint of discrimination the business organization being accused will be notified and given the opportunity to rectify the situation.
In the case of discrimination retail organizations who have over a certain number of employees must comply. For this reason some of these laws may not apply to single-store retailers or retailers with a smaller organizational structure. Regardless of the size or type of retail store, management’s daily operations when dealing with subordinates such as hiring, firing, promotions and raises should be carried out in the most ethical manner.
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https://www.law.cornell.edu/wex/equal_employment_opportunity_commission
↵
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https://www.dol.gov/general/topic/discrimination/agedisc
↵
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https://www.eeoc.gov/employees/charge.cfm
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https://www.eeoc.gov/eeoc/statistics/enforcement/index.cfm
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https://www.nolo.com/legal-encyclopedia/disparate-treatment-discrimination.html
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Contributors and Attributions
- Equal Employment Opportunity Law. Authored by : Freedom Learning Group. Provided by : Lumen Learning. License : CC BY: Attribution


10.2.1: Let's talk about sex!
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- Last updated
- Save as PDF
Did that subtitle catch your attention? Good! But in this section we won’t be discussing intercourse, or sexual activity; rather, we’ll be talking about sex ( as a biological category) and gender. People often use the terms “sex” and “gender” interchangeably; however, these terms are two separate entities.  Every egg is an X chromosome, and sperm can be either X or Y chromosome. Therefore, if an X sperm reaches the egg, the individual will have XX chromosomes (female), and if a Y sperm reaches the egg, the individual will have XY chromosomes (male). The term “sex” refers to biology and should be used when biological distinctions are emphasized. For example, “There are sex differences in hormone production”.
The main differences between sexes are reproductive body parts, which develop in reaction to hormone levels introduced when the embryo is only about nine weeks. At this point in the pregnancy, the embryo will develop the gonads, or reproductive glands. They are the tiny beginnings of either testes or ovaries. However, hormonal anomalies can (and do) occur during those early stages of development. These anomalies can result in external genitalia not being easily discernable as either male or female. In these cases, the individual will likely be identified as intersexed .
Males and females have far more in common than they have differences. For example, think about our organs, hair, skin, limbs, nervous systems, endocrine systems, etc. The similarities in these things far outweigh the differences in males and females. However, we pay a lot of social attention to the differences between sexes, often even looking to those physiological differences to explain behavioral differences.
Gender , on the other hand, serves as a cultural indicator of a person’s personal and social identity. 8 Gender, for the individual, starts with sex assignment, and sex is assigned on the basis of the observable genitalia at birth. In other words, sex is nature, and gender is nurture. Sex is predetermined, and gender is learned through socialization. Gender is something we do . 9 Therefore, gender is a psychologically ingrained social construct that actively surfaces in everyday human interaction and behaviors. We learn masculinity; we learn femininity. Then we do masculinity, and we do femininity. And how about all those behaviors we don’t describe as either strictly masculine or feminine? Enter androgyny ! Androgynous behaviors are all behaviors that do not fit neatly into our constructions of masculine or feminine. Therefore, our behaviors can be described as masculine, feminine, or androgynous. And just like walking or talking are learned behaviors, so are our interpretations of labeling our behaviors as masculine, feminine, or androgynous.
Gender refers to the attitudes, feelings, and behaviors that a given culture associates with a person’s biological sex. Behavior that is compatible with cultural expectations is referred to as gender-normative ; behaviors that are viewed as incompatible with these expectations constitute gender non-conformity . 10 Gender expression is the presentation of an individual, including physical appearance, clothing choice and accessories, and behaviors that express aspects of gender identity or role. Gender expression may or may not conform to a person’s gender identity. Gender identity refers to one’s sense of oneself as masculine, feminine, androgynous, or transgender. 11 Gender dysphoria refers to discomfort or distress that is associated with a discrepancy between a person's gender identity and that person's sex assigned at birth. 12 Transgenderis an umbrella term used to describe the full range of people whose gender identity and/or gender role do not conform to what is typically associated with their sex assigned at birth.13
Gender is shaped through our learning process, and is influenced by all kinds of things outside of the individual, such as family, friends, television, social media, teachers, politicians, legislation, institutions, and culture. Western gender conceptualization reflects a bipolar construction, recognizing two major gender categories (masculinity and femininity) as being dominant. Western construction of gender also reinforces the idea that males are masculine while females are feminine, and often this idea is taken further with the western interpretation of gender suggesting people are born a specific gender. Constricting? Yes. Over-simplified? Absolutely. Completely inaccurate? You betcha’! This interpretation of gender is narrowly fashioned by a traditional European perpesctive. Take, for instance, many of the Plains Indians’ interpretation of gender wherein people can be two-spirited , known as berdache. 14 Berdache was not recognized as a third category; instead, the term was used to describe the continuum of human behavior that didn’t fit neatly into European notions of what it meant to be male or female. In fact, the term berdache has a history deflecting its Eurocentric origins and ethnocentrism of most 17th, 18th, and 19th century European and European American observers of Native American cultures.
Identical male twins, Bruce and Brian Reimer, were born in 1963 to parents Janet and Ron Reimer.  Janet Reimer wrote to Dr. John Money seeking help for her son who she feared would not have a sex or gender identity because his genitalia had been mutilated. At the strong encouragement of Dr. Money, the boy’s parents opted for sex reassignment surgery for their 17-month-old baby. The boy was sexually reassigned to a female with lifelong hormone therapy. Dr. Money declared the surgery and reassignment a total success in 1973. From a young age, Bruce (who had been renamed Brenda) demonstrated behaviors indicating she did not want to wear dresses, play with dolls, or help with traditionally feminine household chores. At age 12 (many years earlier) Brenda began experiencing severe emotional problems because, even though she was receiving estrogen regularly, she appeared very masculine and was the brunt of a lot of joking and bullying in school and from her peers. At 14 years old, the child refused any more hormone therapy or genital surgeries, and her father finally told her the truth about her medical history. She was reportedly relieved and started male hormone therapy, underwent genital reconstruction surgery to create a penis, and had a mastectomy, and changed his name to David. Researchers concluded a person could be “successfully” socialized as a female because essentially his brain “knew otherwise.”
In addition to his difficult lifelong relationship with his parents and peers and his own identity, Reimer had to deal with unemployment and the death of his brother Brian from an overdose of antidepressants in 2002. In 2004, his wife Jane told him she wanted to separate. On the morning of May 4, 2004, Reimer drove to a grocery store's parking lot and took his own life by shooting himself in the head with a sawed- off shotgun. He was 38 years old.
David Reimer taught us a lot about sex category and gender attainment. While gender is learned through a socialization process, we cannot socialize or train someone into their sex identity. In other words, a boy can be socialized to “act like a girl” or a girl can be socialized to “act like a boy,” but you cannot socialize someone into believing they are the opposite sex. He also taught us our brains develop as being either male or female. And most often time, our genital development with align with our brain’s development, but sometimes not. David Reimer reported that from a young age he “just knew” he wasn’t a girl, even though he never knew he had been born a male.
Testosterone and Gender
Males and females produce the same hormones, but in different amounts. Females secrete more estrogen and males secrete more testosterone. Studies conducted have, in fact, concluded more testosterone does lead to more aggressive behavior. However most of these studies have been done in animals, wherein the animal will be injected with high levels of testosterone and aggressive behavior will ensue. Many people feel these findings are indicative of male behaviors, sometimes been excusing overly aggressive behavior with excuses like, “Boys will be boys.” But maybe a more appropriate response would be “boys will be a$$h*les” when we see overly aggressive behavior. Because people need to be cautious when interpreting results from the previously mentioned studies, as there is tremendous variation in behavior among animal species.
In research in human, higher levels of testosterone have been linked to higher levels of edginess, competitiveness, and anger—in females and males. Also, hormone levels fluctuate throughout the day and are influenced by environment. Think about it: What do you think your hormone levels look like when you’re about to jump out of a plane versus sitting on the couch watching TV? Therefore, biology itself is influenced and affected by environment. The myth that testosterone alone affects men’s behavior has been debunked.
In addition, hormones do not dictate behavior. Hormones may affect mood, but they do not dictate behavior. Consider a time when you felt like you were unable to demonstrate your desired behaviors because of the social context in which you were engaged. Maybe you were in a classroom or a workplace or whatever, and when you were feeling aggressive, upset, or angry you were able to control your emotions and behaviors because of the social context. So while hormones were affecting mood, they did not dictate your behavior. You were able to control your behavior because of the setting you were in. Therefore, behavior is highly governed, not by hormone, but by the situation or context in which it occurs.
Further, research indicates women can be just as aggressive as men when they’re either rewarded for their behaviors (athletes) or when they think it is safe to do so while avoiding social sanctions. Think about people like Ronda Rousey, Serena Williams, Hillary Clinton, Pink, or Chyna. All of these women have in common being rewarded for what have traditionally been described as masculine behaviors. Is there something wrong with them? No. Are they biologically less female than other females? No.
Women, Hormones, and Behaviors
Jokes about women’s hormone levels are old ones. Seriously, very old. Like, we need some new material. How are people even still laughing at PMS jokes? I don’t know, but women’s hormone levels (especially testosterone) do not fluctuate as much as men’s throughout the day. I’m going to say that again to make sure you really read that last sentence. Women’s hormone levels (especially testosterone) do not fluctuate as much as men’s throughout the day. Rather female hormone production is influenced by the monthly reproductive cycles as well as the cessation of the reproductive capacity (menopause) during the life course. So, really, instead of accusing women of getting crazy once a month, women could accuse men of getting crazy every day! No, no, I’m kidding, and that is not the kind of equality sociologists are fighting for.
Both girls and boys learn negative attitudes toward menstruation at early ages. These negative beliefs at influence women’s experiences with PMS symptoms. Meaning, if women are taught PMS id horrible, women may expect the horridness, and then produce behaviors reflective of that horridness. While many women report mood wings, research shows that negative chance in mood as well as physical changes mat nave more to do with stressful external events than with the phase of PMS. So, women might be reacting to PMS and excusing our sometimes horrid behavior by blaming it on this occurring phase. Kind of like that whole “boys will be boys” thing, huh?
In studies where both men and women participated, men were equally likely to express mood swings, problems at work, and physical discomfort. Men are actually subject to a daily hormone cycle in which testosterone levels peak at about 4am and are lowest at 8pm. “When people say women can’t be trusted because they cycle every month, my response is that men cycle every day, so they should only be allowed to negotiate peace treaties in the evening.” -- June Reinisch.
7
Young, R. (20090401). Sex/Gender.
AMA Manual of Style
. Retrieved 17 Oct. 2016, from
http://www.amamanualofstyle.com/view
/10.1093/jama/9780195176339.001.0001/med-9780195176339-div2-350.
8
Ibid
9
Doing Gender
. Candace West; Don H.
Zimmerman
. Gender and Society, Vol. 1, No. 2. (Jun., 1987), pp. 125-151.
10
American Psychological Association. (2012). Guidelines for Psychological Practice with Lesbian, Gay, and Bisexual
Clients. American Psychologist, 67(1), 10–42. doi: 10.1037/a0024659
11
Ibid
12
American Psychological Association & National Association of School Psychologists. (2015). Resolution on gender and
sexual orientation diversity in children and adolescents in schools. Retrieved from
http://www.apa.org/about/policy/orie...diversity.aspx
13
American Psychological Association. (2015). Guidelines for Psychological Practice with Transgender and Gender
Nonconforming People. American Psychologist, 70(9), 832-864.
14
Estrada, Gabriel S. 2011. "Two Spirits, Nádleeh, and LGBTQ2 Navajo Gaze."
American Indian Culture and Research
Journal
35(4):167-190.


9.3: Pierrot Lunaire
Pierrot Lunaire is a song cycle. It is written in three parts with each part containing seven songs. The piece on our playlist, “Madonna,” is song number 6 from Part One. It was composed during Schoenberg’s second period after the composer had turned to atonality but before he developed his twelve-tone method. The inward psychological focus of the text and the eerie combination of atonality and sprechstimme mark this as a clearly expressionist work. Sprechstimme is an expressionist technique in which the singer performs the musical line in a half-sung, half-spoken style. The written notes on the page are used as a guide but are only approximated by the singer.
Introduction
Dreimal sieben Gedichte aus Albert Girauds “Pierrot lunaire” (“Three times Seven Poems from Albert Giraud’s ‘Pierrot lunaire'”), commonly known simply as Pierrot Lunaire , Op. 21 (“Moonstruck Pierrot” or “Pierrot in the Moonlight”), is a melodrama by Arnold Schoenberg. It is a setting of 21 selected poems from Otto Erich Hartleben’s German translation of Albert Giraud’s cycle of French poems of the same name. The première of the work, which is between 35 and 40 minutes in length, was at the Berlin Choralion-Saal on October 16, 1912, with Albertine Zehme as the vocalist.
The narrator (voice-type unspecified in the score, but traditionally performed by a soprano) delivers the poems in the Sprechstimme style. Schoenberg had previously used a combination of spoken text with instrumental accompaniment, called “melodrama,” in the summer-wind narrative of the Gurre-Lieder , and it was a genre much in vogue at the end of the nineteenth century. The work is atonal but does not use the twelve-tone technique that Schoenberg would devise eight years later.
History
The work originated in a commission by Zehme for a cycle for voice and piano, setting a series of poems by the Belgian writer Albert Giraud. The verses had been first published in 1884, and later translated into German by Otto Erich Hartleben. Schoenberg began on March 12 and completed the work on July 9, 1912, having expanded the forces to an ensemble consisting of flute (doubling on a piccolo), clarinet (doubling on bass clarinet), violin (doubling onviola), cello, and piano. After forty rehearsals, Schoenberg and Zehme (in Columbine dress) gave the premiere at the Berlin Choralion-Saal on October 16, 1912. Reaction was mixed. According to Anton Webern, some in the audience were whistling and laughing, but in the end “it was an unqualified success.”  Instead, they would go away whistling the tunes.” The show took to the road throughout Germany and Austria later in 1912. It was performed for the first time in the western hemisphere at the Klaw Theatre in New York City on February 4, 1923, with George Gershwin and Carl Ruggles in attendance.
Structure
“Pierrot Lunaire” consists of three groups of seven poems. In the first group, Pierrot sings of love, sex and religion; in the second, of violence, crime, and blasphemy; and in the third of his return home to Bergamo, with his past haunting him.
| Part One | Part Two | Part Three |
|---|---|---|
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|
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Schoenberg, who was fascinated by numerology, also makes great use of seven-note motifs throughout the work, while the ensemble (with conductor) comprises seven people. The piece is his opus 21, contains 21 poems, and was begun on March 12, 1912. Other key numbers in the work are three and 13: each poem consists of 13 lines (two four-line verses followed by a five-line verse), while the first line of each poem occurs three times (being repeated as lines seven and 13).
Music
Pierrot Lunaire uses a variety of classical forms and techniques, including canon, fugue, rondo, passacaglia and free counterpoint. The poetry is a German version of a rondeau of the old French type with a double refrain. Each poem consists of three stanzas of 4 + 4 + 5 lines, with line 1 a Refrain (A) repeated as line 7 and line 13, and line 2 a second Refrain (B) repeated for line 8.
The instrumental combinations (including doublings) vary between most movements. The entire ensemble plays together only in the 11th, 14th and final 4 settings.
The atonal, expressionistic settings of the text, with their echoes of German cabaret, bring the poems vividly to life. Sprechgesang , literally “speech-singing” in German, is a style in which the vocalist uses the specified rhythms and pitches, but does not sustain the pitches, allowing them to drop or rise, in the manner of speech.
Analysis
Pierrot Lunaire is a work which can be interpreted through the sixth song “Madonna”. In this song the only person who could save Pierrot, Jesus, is presented as dead. After a brief period of sorrow in “Der kranke Mond” Pierrot in Part II of the song cycle becomes more depraved in his exploits and by the end is crucified for his sins in “Die Kreuze.” Hoping to redeem himself in Part III, Pierrot tries to go back to previous persona as the “old pantomime from Italy” but ultimately fails without much hope of redemption by the end of the work.
Nacht
Pitch Structures
Everything in “Nacht” is generated from a ten-note motif, introduced in canon starting in the fourth bar. This piece predominantly uses the pitch collection (014). This collection gets introduced in the very first measure with the piano. If you take into account every note the piano plays in the first three measures you get an octatonic scale (0134679T). This is just four groups of (014) transposed by T3. In measure 4, we see one set of (014) in the bass clarinet line. Whenever we see this pitch collection it is usually in one of two rhythms. The first, as illustrated by the bass clarinet in measure 4, is three half notes. Later it shows up as three quarter notes, simply a compressed version of the original. The other rhythm is three eighth notes, usually this comes in groups of three. For example, in measure 8 the bass clarinet has the collection three times, each time as an eighth note rest followed by three eighth notes. If you look at that measure as a whole, and any other time this pattern shows up, we see that the second group is transposed from the first by T4. The third group of notes is transposed from the first group by T1. Thus, the entire measure is actually using (014), not just in each group, but within each group. In other words, if you take the first note of each three you get (014), if you take the second note of each you get (014), and if you take the third note you also get (014). For the first two stanzas of text, we only ever see (014) transposed by various Tx. Starting in the third stanza, we begin to see inversions of (014). In measure 19, the right hand in the piano starts off with an inversion of (014) and then goes to a transposition of (014). This continues on for the entirety of the run in both hands of the piano.
Formal Structure
Following a brief introduction, the movement falls into three strophes of seven, six, and seven bars, with section breaks occurring at measures 11 and 17, articulated by a change of tempo to etwas rascher for the second strophe, and back again to the initial tempo for the third. These follow the stanzas of the poem, and are followed by a coda. The first strophe is canonic in four voices; the second is also canonic, but in just three voices; the third strophe consists of a rapid succession of ambiguous canonic fragments.
Although the pitch-class sets are virtually the same throughout the whole movement, each section has distinct musical elements that differentiate them from each other. When looking at the transition from the first section to the second section, a couple important changes take place. First, the tempo marking “Etwas rascher” marks an increase in speed and rhythmic energy. This is further exaggerated by the increased rhythmic density in the piano, clarinet, and cello parts. In addition to the rhythmic changes, the registers of the cello, piano, and vocal lines are notably higher in the second section. The third section returns to the original tempo, and the register and rhythmic density change again to closely resemble the first section. This analysis provides evidence for organizing the piece into an ABA’ form, as the first and last sections have many similar elements while the middle section differs substantially. However, because of the pitch-class set similarities, the argument could be made for a quasi theme-and-variation organization, with A, A’, and A sections.
Musical Structure to Extramusical Elements
Expressionism is a modernist movement that began in Germany and Austria in the early twentieth century. Schoenberg, Austrian in descent, was associated with the expressionist movement in German poetry and art. Being from Austria at this time, his music was often labeled as degenerate music since Schoenberg is Jewish. Expressionistic music is dominated by dissonance rather than consonance, and can create an “unsettling” feeling among its listeners For many, expressionistic music meant a rejection of the past and an acceptance of the innovative, unfamiliar future. The text of “Nacht” can be described as ominous, and depicts the wings of black moths covering the sun. These views are characteristic of expressionistic poetry.
“Nacht” also employs the occasional use of word painting through his music where he uses the music to illustrate the literal meaning of a particular word [4]. These text expressions make general associations between the text and musical setting. This can be seen with the word “duft” translating to “scent” in measure 12. The full poetic phrase “Steigt ein Duft” means “arises a scent,” and this is depicted by a leap upwards in voice from A to G ♭ . Likewise, the word “verschwiegen” loosely translates to “mutely” or “hushed”, and is performed by conventional singing rather than Sprechstimme , along with prolonged silence with a fermata, though the word more precisely means “discreetly” or “closed-mouthed.”
Notable Recordings
Notable recordings of this composition include:
| Sprechstimme | Ensemble | Conductor | Record Company | Year of Recording | Format |
|---|---|---|---|---|---|
| Erika Stiedry-Wagner | Arnold Schoenberg | Columbia Records | 1940 | n/a [11] | |
| Helga Pilarczyk | Members of the Conservatory Society Concert Orchestra | Pierre Boulez | Ades | 1961 | CD |
| Bethany Beardslee | Columbia Chamber Ensemble | Robert Craft | Columbia / CBS | 1963 | CD |
| Jan DeGaetani | Contemporary Chamber Ensemble | Arthur Weisberg | Nonesuch | 1970 | CD |
| Yvonne Minton | Ensemble InterContemporain | Pierre Boulez | Sony Music | 1977 | CD |
| Barbara Sukowa | Schoenberg Ensemble | Reinbert de Leeuw | Koch Schwann | 1988 | CD |
| Jane Manning | Nash Ensemble | Simon Rattle | Chandos | 1991 | CD |
| Phyllis Bryn-Julson | Ensemble Modern | n/a | BMG | 1991 | CD |
| Phyllis Bryn-Julson | New York New Music Ensemble | Robert Black | GM Recordings | 1992 | CD |
| Karin Ott | Cremona Musica Insieme | Pietro Antonini | Nuova Era | 1994 | CD |
| Christine Schäfer | Ensemble InterContemporain | Pierre Boulez | Deutsche Grammophon | 1997 | CD [12] |
| Anja Silja | Twentieth Century Classics Ensemble | Robert Craft | Naxos | 1999 | CD [13] |
Arnold Schoenberg himself made test recordings of the music with a group of Los Angeles musicians from September 24 to 26, 1940. These recordings were eventually released on LP by Columbia Records in 1949, and reissued in 1974 on the Odyssey label.
The avant-pop star Björk, known for her interest in avant-garde music, performed Pierrot Lunaire at the 1996 Verbier Festival with Kent Naganoconducting. According to the singer in a 2004 interview, “Kent Nagano wanted to make a recording of it, but I really felt that I would be invading the territory of people who sing this for a lifetime.” Only small recorded excerpts (possibly bootlegs) of her performance have become available.
The jazz singer Cleo Laine recorded Pierrot Lunaire in 1974. Her version was nominated for a classical Grammy Award. Another jazz singer who has performed the piece is Sofia Jernberg, who sang it with Norrbotten NEO.
In March 2011, Bruce LaBruce directed a performance at the Hebbel am Ufer Theatre in Berlin. This interpretation of the work included gender diversity, castration scenes and dildos, as well as a female to male transgender Pierrot. LaBruce subsequently filmed this adaptation as the 2014 theatrical film Pierrot Lunaire .
Legacy as a Standard Ensemble
The quintet of instruments used in Pierrot Lunaire became the core ensemble for The Fires of London, who formed in 1965 as “The Pierrot Players” to perform Pierrot Lunaire , and continued to concertize with a varied classical and contemporary repertory. This group performed works arranged for these instruments and commissioned new works especially to take advantage of this ensemble’s instrumental colors, up until it disbanded in 1987.
Over the years, other groups have continued to use this instrumentation professionally (current groups include Da Capo Chamber Players, eighth blackbird) and the Finnish contemporary group Uusinta Lunaire, and have built a large repertoire for the ensemble.
Contributors and Attributions
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution
- Pierrot Lunaire. Provided by : Wikipedia. Located at : http://en.Wikipedia.org/wiki/Pierrot_Lunaire . License : CC BY-SA: Attribution-ShareAlike


10.1: Vocabulario. La ropa
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- Romano Sánchez Domínguez
- Imperial Valley College
Objetivos
At the end of this module, students will be able to:
- Identify the new vocabulary.
- Recognize the pronunciation of the words.
- Listen and match with the appropriate words.
Vocabulario: La ropa
Un verbo útil ( a useful verb ):
Llevar ( to wear; also means to carry and to bring )
Text only list
Las prendas de ropa (articles of clothing)
- el abrigo (coat)
- el bolso (bag / purse)
- las botas (boots)
- la bufanda (scarf)
- los calcetines (socks)
- la camisa (shirt)
- la camiseta (t-shirt)
- la chaqueta (jacket)
- la falda (skirt)
- la gorra (baseball cap)
- el gorro (hat)
- los guantes (gloves)
- el impermeable (raincoat)
- los lentes (eyeglasses)
- los pantalones (pants / trousers)
- los pantalones cortos (shorts)
- el paraguas (umbrella)
- el pijama (pyjamas)
- el sombrero (hat)
- el suéter (sweater)
- los tenis (sneakers)
- el traje (suit)
- el traje de baño (swimsuit)
- el vestido (dress)
- los zapatos (shoes)
Un verbo útil (a useful verb):
-
Llevar (to wear; also means to carry and to bring)
Contributors and Attributions
CC licensed content, Shared previously
- Mens Slim Fit Designer Casual Trench Coat. Authored by : FLATSEVEN. Provided by : Flickr. Located at : https://www.flickr.com/photos/flatseven/35451357861 . License : CC BY-SA: Attribution-ShareAlike
- Botas de Trabajo. Authored by : pen_ash. Provided by : Pixabay. Located at : https://pixabay.com/es/photos/botas-de-trabajo-botas-los-zapatos-4133817/ . License : Other . License Terms : Pixabay License
- Bufanda India Colores. Authored by : janin_amandine. Provided by : Pixabay. Located at : https://pixabay.com/es/photos/bufanda-india-colores-colorido-1442583/ . License : Other . License Terms : Pixabay License
- Pies Calcetines. Authored by : tookapic. Provided by : Pixabay. Located at : https://pixabay.com/es/photos/pies-calcetines-persona-piso-933087/ . License : Other . License Terms : Pixabay License
- En Blanco Camisetas. Authored by : nattyjeff. Provided by : Pixabay. Located at : https://pixabay.com/es/illustrations/en-blanco-camisetas-blanco-1976334/ . License : Other . License Terms : Pixabay License
- chaqueta. Authored by : StockSnap. Provided by : Pixabay. Located at : https://pixabay.com/es/photos/azul-mezclilla-chaqueta-2566082/ . License : Other . License Terms : Pixabay License
- Falda Moda. Authored by : Engin_Akyurt. Provided by : Pixabay. Located at : https://pixabay.com/es/photos/falda-moda-pierna-ni%C3%B1a-mujer-sexy-3897351/ . License : Other . License Terms : Pixabay License
- Gorro de Ochos. Authored by : Merche Lazaro. Provided by : Flickr. Located at : https://www.flickr.com/photos/deunvistazo/5502617442/ . License : CC BY: Attribution
- Guantes El Dedo. Authored by : Engin_Akyurt. Provided by : Pixabay. Located at : https://pixabay.com/es/photos/guantes-el-dedo-letras-msn-2685978/ . License : Other . License Terms : Pixabay License
- Niu00f1o Niu00f1a. Authored by : mintchipdesigns. Provided by : Pixabay. Located at : https://pixabay.com/es/photos/ni%C3%B1o-ni%C3%B1a-la-lluvia-puddle-504321/ . License : Other . License Terms : Pixabay License
- lentes. Authored by : yankara. Provided by : Flickr. Located at : https://www.flickr.com/photos/yankara/473075608/ . License : CC BY: Attribution
- Jeans Pantalones. Authored by : Tama66. Provided by : Pixabay. Located at : https://pixabay.com/es/photos/jeans-pantalones-estilo-hombre-4050815/ . License : Other . License Terms : Pixabay License
- Men's Shorts - Old Bull Lee. Authored by : OldBullLeeWasHere. Provided by : Wikimedia Commons. Located at : https://commons.wikimedia.org/wiki/File:Men%27s_Shorts_-_Old_Bull_Lee.jpg . License : CC BY-SA: Attribution-ShareAlike
- Child K's Christmas pajamas. Authored by : Fresh On The Net. Provided by : Flickr. Located at : https://www.flickr.com/photos/feelingmyage/6574066213 . License : CC BY: Attribution
- Sombrero. Authored by : Gellinger. Provided by : Pixabay. Located at : https://pixabay.com/es/photos/sombrero-sombrero-de-paja-headwear-2519955/ . License : Other . License Terms : Pixabay License
- Suu00e9ter Gris. Authored by : Engin_Akyurt. Provided by : Pixabay. Located at : https://pixabay.com/es/photos/su%C3%A9ter-gris-hombre-invierno-fr%C3%ADo-3737417/ . License : Other . License Terms : Pixabay License
- Siblings?. Authored by : Chris Hunkeler. Provided by : Flickr. Located at : https://www.flickr.com/photos/chrishunkeler/42040859702 . License : CC BY-SA: Attribution-ShareAlike
- Evening Dresses. Authored by : Leopold Terence. Provided by : Flickr. Located at : https://www.flickr.com/photos/98794605@N03/13159956955/ . License : CC BY-SA: Attribution-ShareAlike
- Zapatos Negros. Authored by : xaytham. Provided by : Pixabay. Located at : https://pixabay.com/es/photos/zapatos-negros-zapatos-de-cuero-2752226/ . License : Other . License Terms : Pixabay License
- Cap. Authored by : Marcelo. Located at : https://pixnio.com/objects/shoes-and-clothing/white-sports-cap-hat# . License : CC0: No Rights Reserved
Public domain content
- Bolso. Located at : https://pxhere.com/es/photo/673931 . License : CC0: No Rights Reserved
- Silk Hawaiian shirt, Hawaiian Surf on label, HSTR 17416.. Provided by : National Parks Service. Located at : https://www.nps.gov/hstr/learn/historyculture/the-casual-sport-shirt.htm . License : Public Domain: No Known Copyright
- Calle Caminando Paraguas Persona. Authored by : Nayara Sabino. Provided by : Pexels. Located at : https://www.pexels.com/es-es/foto/calle-caminando-paraguas-persona-1844746/ . License : Public Domain: No Known Copyright
- Zapatos. Located at : https://pxhere.com/es/photo/1207081 . License : CC0: No Rights Reserved
- Man Wearing Suit Jacket. Authored by : Walid Riachy. Located at : https://www.pexels.com/photo/man-wearing-suit-jacket-dress-shirt-and-dress-pants-standing-near-wall-1450114/ . License : Public Domain: No Known Copyright
- Vocabulario: La ropa. Authored by : SUNY Oneonta with Lumen Learning. Provided by : SUNY Oneonta. License : CC BY: Attribution


2.2: Streak Plate Technique Procedure
Quadrant Streak Plate Procedure
Name: _____________________________________________________
Course Section: ______________________________________________
The following activity will teach students how to complete a Quadrant Streak with the goal of generating isolated, pure colonies for colony morphology. Read the following procedure carefully before starting the activity
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Procedure |
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Students will select 1 bacterial species and work alone for this experiment. After incubation, students will compare results with classmates to compete the results chart.
Step 1: Label the bottom of the nutrient agar plate with student initials, course section, date, and the bacteria selected.
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Step 2: Heat an inoculation loop in the incinerator for at least 10 seconds until the wire is red/orange.
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Procedure |
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Step 3: Once the loop is cool, aseptically collect 1 colony from the stock plate. You do not need to swipe or gather a large amount of bacteria.
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Step 4: Sterilize the loop again and allow it to cool. Then, place the loop on a small portion of the quadrant 1 streak.
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Step 5: Sterilize the loop again and repeat Step 4, this time dragging bacteria from quadrant 2 into quadrant 3.
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Results
In table below, record the 4 colony morphology characteristic listed for your streak plate. Then, compare results with 3 other classmates to fill in the morphologies for the remaining bacterial species.
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| Bacterial Species | General Shape and Color | Colony Margin | Colony Elevation | Colony Size |
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Escherichia coli
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Serratia marcescens
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Micrococcus luteus
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Staphylococcus epidermidis
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General Questions
1. Which bacterial species did you use for your streak plate? Did you successfully obtain isolated colonies?
2. What are potential errors that may result in a failed streak plate?
3. Explain the importance of the streak plate technique in microbiology.
4. Based on proper streak plate technique, which area of the plate will always have:
- The most microbial growth:
- The least microbial growth:
5. Why can isolated, single colonies be used to generate pure cultures?
Health-Related Questions
1. How are microbiology streak plate and culture techniques useful for diagnosing strep throat?
2. A patient went to the doctor's office for a sore throat and fever. The doctor took a throat swab of the patient for a culture test, but no bacteria grew. Is Streptococcus pyogenes the causative agent of this infection? Why or why not?
Attributions
"Microbiology Laboratory Manual: Labs, 1.8 Plating on Petri Plates for Isolation" by Dr. Rosanna Hartline , LibreTexts: Biology , West Hills College Lemoore is licensed under CC BY-NC-SA 4.0
Image Citations
Procedure Image, The 4 streak plate images used in the procedure table were modified from:
- Streak plates 1.svg by Reytan, Wikimedia Commons is in the Public Domain
- Streak plates 2.svg by Reytan, Wikimedia Commons is in the Public Domain
- Streak plates 3.svg by Reytan, Wikimedia Commons is in the Public Domain
- Streak plates 4.svg by Reytan, Wikimedia Commons is in the Public Domain


6: Growing Diversity of Plants
When plants developed basic tissues and organs and thus became mature enough to survive on land, they started to increase in their diversity. All plants studied in this and following chapters belong to plants\(_2\), or kingdom Vegetabilia which is split into three phyla (Figure 6.1.1): Bryophyta (mosses and relatives), Pteridophyta (ferns and allies), and Spermatophyta (seed plants). The most striking differences between these phyla lay in the organization of their life cycles.
Land plants have a sporic life-cycle (Figure 4.4.7) that begins with a diplont (sporophyte); the mother cell of spores goes through the meiosis and produces haploid spores. These spores develop into haplont which produces female and male gametangia (gamete “homes”). Female is called archegonium , the male— antheridium ; the archegonium produces oocyte which is fertilized by the antheridium’s spermatozoon in the process of oogamy. When this fertilization happens, it forms a diploid zygote which then matures into a young sporophyte growing on a gametophyte . This kind of same species parasitism is almost unique in the living world. Only viviparous animals (like mammals with their pregnancy) could be compared with land plants.
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- 6.1: Bryophyta - the Mosses
- Bryophyta has approximately 20,000 species. They do not have roots, but have long dead cells capable of water absorbency via apoplastic transport, these cells are called rhizoid cells. Their sporophyte is reduced to sporogon, which is simply a sporangium with setamosses: stalk of the sporogon (see) (stalk), and is usually parasitic. Gametophyte of bryophytes starts its development from a protonema, thread of cells.
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- 6.2: Pteridophyta - the Ferns
- Pteridophyta, ferns and allies, have approximately 12,000 species and six classes. They have a sporic life cycle with sporophyte predominance whereas their gametophytes are often reduced to prothallium, small hornwort-like plant. Another frequent variant is the underground, mycoparasitic gametophyte. Pteridophyta (with one exception) have true roots. Most of them have vascular tissues and are homoiohydric. This is why seed plants together with ferns have a name vascular plants.


3.1: Descartes’ Meditations - Cartesian Dualism
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Meditations 11
MEDITATION VI- OF THE EXISTENCE OF MATERIAL THINGS, AND OF THE REAL DISTINCTION BETWEEN THE MIND AND BODY OF MAN.
1. THERE now only remains the inquiry as to whether material things exist. With regard to this question, I at least know with certainty that such things may exist, in as far as they constitute the object of the pure mathematics, since, regarding them in this aspect, I can conceive them clearly and distinctly. For there can be no doubt that God possesses the power of producing all the objects I am able distinctly to conceive, and I never considered anything impossible to him, unless when I experienced a contradiction in the attempt to conceive it aright. Further, the faculty of imagination which I possess, and of which I am conscious that I make use when I apply myself to the consideration of material things, is sufficient to persuade me of their existence: for, when I attentively consider what imagination is, I find that it is simply a certain application of the cognitive faculty ( facultas cognoscitiva) to a body which is immediately present to it, and which therefore exists.
2. And to render this quite clear, I remark, in the first place, the difference that subsists between imagination and pure intellection [or conception ]. For example, when I imagine a triangle I not only conceive (intelligo) that it is a figure comprehended by three lines, but at the same time also I look upon (intueor) these three lines as present by the power and internal application of my mind (acie mentis), and this is what I call imagining. But if I desire to think of a chiliogon, I indeed rightly conceive that it is a figure composed of a thousand sides, as easily as I conceive that a triangle is a figure composed of only three sides; but I cannot imagine the thousand sides of a chiliogon as I do the three sides of a triangle, nor, so to speak, view them as present [with the eyes of my mind ]. And although, in accordance with the habit I have of always imagining something when I think of corporeal things, it may happen that, in conceiving a chiliogon, I confusedly represent some figure to myself, yet it is quite evident that this is not a chiliogon, since it in no wise differs from that which I would represent to myself, if I were to think of a myriogon, or any other figure of many sides; nor would this representation be of any use in discovering and unfolding the properties that constitute the difference between a chiliogon and other polygons. But if the question turns on a pentagon, it is quite true that I can conceive its figure, as well as that of a chiliogon, without the aid of imagination; but I can likewise imagine it by applying the attention of my mind to its five sides, and at the same time to the area which they contain. Thus I observe that a special effort of mind is necessary to the act of imagination, which is not required to conceiving or understanding (ad intelligendum); and this special exertion of mind clearly shows the difference between imagination and pure intellection (imaginatio et intellectio pura).
3. I remark, besides, that this power of imagination which I possess, in as far as it differs from the power of conceiving, is in no way necessary to my [nature or] essence, that is, to the essence of my mind; for although I did not possess it, I should still remain the same that I now am, from which it seems we may conclude that it depends on something different from the mind. And I easily understand that, if some body exists, with which my mind is so conjoined and united as to be able, as it were, to consider it when it chooses, it may thus imagine corporeal objects; so that this mode of thinking differs from pure intellection only in this respect, that the mind in conceiving turns in some way upon itself, and considers some one of the ideas it possesses within itself; but in imagining it turns toward the body, and contemplates in it some object conformed to the idea which it either of itself conceived or apprehended by sense. I easily understand, I say, that imagination may be thus formed, if it is true that there are bodies; and because I find no other obvious mode of explaining it, I thence, with probability, conjecture that they exist, but only with probability; and although I carefully examine all things, nevertheless I do not find that, from the distinct idea of corporeal nature I have in my imagination, I can necessarily infer the existence of any body.
4. But I am accustomed to imagine many other objects besides that corporeal nature which is the object of the pure mathematics, as, for example, colors, sounds, tastes, pain, and the like, although with less distinctness; and, inasmuch as I perceive these objects much better by the senses, through the medium of which and of memory, they seem to have reached the imagination, I believe that, in order the more advantageously to examine them, it is proper I should at the same time examine what sense-perception is, and inquire whether from those ideas that are apprehended by this mode of thinking ( consciousness), I cannot obtain a certain proof of the existence of corporeal objects.
5. And, in the first place, I will recall to my mind the things I have hitherto held as true, because perceived by the senses, and the foundations upon which my belief in their truth rested; I will, in the second place, examine the reasons that afterward constrained me to doubt of them; and, finally, I will consider what of them I ought now to believe.
6. Firstly, then, I perceived that I had a head, hands, feet and other members composing that body which I considered as part, or perhaps even as the whole, of myself. I perceived further, that that body was placed among many others, by which it was capable of being affected in diverse ways, both beneficial and hurtful; and what was beneficial I remarked by a certain sensation of pleasure, and what was hurtful by a sensation of pain. And besides this pleasure and pain, I was likewise conscious of hunger, thirst, and other appetites, as well as certain corporeal inclinations toward joy, sadness, anger, and similar passions. And, out of myself, besides the extension, figure, and motions of bodies, I likewise perceived in them hardness, heat, and the other tactile qualities, and, in addition, light, colors, odors, tastes, and sounds, the variety of which gave me the means of distinguishing the sky, the earth, the sea, and generally all the other bodies, from one another. And certainly, considering the ideas of all these qualities, which were presented to my mind, and which alone I properly and immediately perceived, it was not without reason that I thought I perceived certain objects wholly different from my thought, namely, bodies from which those ideas proceeded; for I was conscious that the ideas were presented to me without my consent being required, so that I could not perceive any object, however desirous I might be, unless it were present to the organ of sense; and it was wholly out of my power not to perceive it when it was thus present. And because the ideas I perceived by the senses were much more lively and clear, and even, in their own way, more distinct than any of those I could of myself frame by meditation, or which I found impressed on my memory, it seemed that they could not have proceeded from myself, and must therefore have been caused in me by some other objects; and as of those objects I had no knowledge beyond what the ideas themselves gave me, nothing was so likely to occur to my mind as the supposition that the objects were similar to the ideas which they caused. And because I recollected also that I had formerly trusted to the senses, rather than to reason, and that the ideas which I myself formed were not so clear as those I perceived by sense, and that they were even for the most part composed of parts of the latter, I was readily persuaded that I had no idea in my intellect which had not formerly passed through the senses. Nor was I altogether wrong in likewise believing that that body which, by a special right, I called my own, pertained to me more properly and strictly than any of the others; for in truth, I could never be separated from it as from other bodies; I felt in it and on account of it all my appetites and affections, and in fine I was affected in its parts by pain and the titillation of pleasure, and not in the parts of the other bodies that were separated from it. But when I inquired into the reason why, from this I know not what sensation of pain, sadness of mind should follow, and why from the sensation of pleasure, joy should arise, or why this indescribable twitching of the stomach, which I call hunger, should put me in mind of taking food, and the parchedness of the throat of drink, and so in other cases, I was unable to give any explanation, unless that I was so taught by nature; for there is assuredly no affinity, at least none that I am able to comprehend, between this irritation of the stomach and the desire of food, any more than between the perception of an object that causes pain and the consciousness of sadness which springs from the perception. And in the same way it seemed to me that all the other judgments I had formed regarding the objects of sense, were dictates of nature; because I remarked that those judgments were formed in me, before I had leisure to weigh and consider the reasons that might constrain me to form them.
7. But, afterward, a wide experience by degrees sapped the faith I had reposed in my senses; for I frequently observed that towers, which at a distance seemed round, appeared square, when more closely viewed, and that colossal figures, raised on the summits of these towers, looked like small statues, when viewed from the bottom of them; and, in other instances without number, I also discovered error in judgments founded on the external senses; and not only in those founded on the external, but even in those that rested on the internal senses; for is there aught more internal than pain ? And yet I have sometimes been informed by parties whose arm or leg had been amputated, that they still occasionally seemed to feel pain in that part of the body which they had lost, --a circumstance that led me to think that I could not be quite certain even that any one of my members was affected when I felt pain in it. And to these grounds of doubt I shortly afterward also added two others of very wide generality: the first of them was that I believed I never perceived anything when awake which I could not occasionally think I also perceived when asleep, and as I do not believe that the ideas I seem to perceive in my sleep proceed from objects external to me, I did not any more observe any ground for believing this of such as I seem to perceive when awake; the second was that since I was as yet ignorant of the author of my being or at least supposed myself to be so, I saw nothing to prevent my having been so constituted by nature as that I should be deceived even in matters that appeared to me to possess the greatest truth. And, with respect to the grounds on which I had before been persuaded of the existence of sensible objects, I had no great difficulty in finding suitable answers to them; for as nature seemed to incline me to many things from which reason made me averse, I thought that I ought not to confide much in its teachings. And although the perceptions of the senses were not dependent on my will, I did not think that I ought on that ground to conclude that they proceeded from things different from myself, since perhaps there might be found in me some faculty, though hitherto unknown to me, which produced them.
8. But now that I begin to know myself better, and to discover more clearly the author of my being, I do not, indeed, think that I ought rashly to admit all which the senses seem to teach, nor, on the other hand, is it my conviction that I ought to doubt in general of their teachings.
9. And, firstly, because I know that all which I clearly and distinctly conceive can be produced by God exactly as I conceive it, it is sufficient that I am able clearly and distinctly to conceive one thing apart from another, in order to be certain that the one is different from the other, seeing they may at least be made to exist separately, by the omnipotence of God; and it matters not by what power this separation is made, in order to be compelled to judge them different; and, therefore, merely because I know with certitude that I exist, and because, in the meantime, I do not observe that aught necessarily belongs to my nature or essence beyond my being a thinking thing, I rightly conclude that my essence consists only in my being a thinking thing [or a substance whose whole essence or nature is merely thinking]. And although I may, or rather, as I will shortly say, although I certainly do possess a body with which I am very closely conjoined; nevertheless, because, on the one hand, I have a clear and distinct idea of myself, in as far as I am only a thinking and unextended thing, and as, on the other hand, I possess a distinct idea of body, in as far as it is only an extended and unthinking thing, it is certain that I, [that is, my mind, by which I am what I am], is entirely and truly distinct from my body, and may exist without it.


7.3: Flow Rate
Flow is the path that a certain liquid (water) takes for a specific time. It is commonly referred to as Flow Rate. A garden hose, a water main, an aqueduct, a gutter, have flow rates. You must know the area of the vessel or container that is flowing the liquid and you must also know the velocity that the liquid is moving. Flow Rate is simply expressed as volume over time.
\[\dfrac{\text { Volume }}{\text { Time }} \nonumber \]
The volume can be gallons, million gallons, cubic feet or any other volume unit. The time can be any time such as, seconds, minutes, hours, etc. However, when solving Flow Rate ALWAYS use the UNTIS – CUBIC FEET PER SECOND, where Q represents Flow Rate.
Expressed as
\[\mathrm{Q}=\dfrac{\text { Volume }}{\text { Time }}=\dfrac{\text { cubic feet }}{\text { second }} \nonumber \]
In order to calculate a Flow Rate, you need to know the Area of the vessel that the liquid is traveling through and the Velocity in which the liquid is traveling. The formula for Flow Rate is
\[\mathrm{Q}=\mathrm{A} \times \mathrm{V} \nonumber \]
Where the Area is expressed as – \(0.785 \times D^2 \) (for circles) - \(L \times H\) (for rectangles) and \(\dfrac{b 1+b 2}{2} \times H\) (for trapezoids)
And
Where Velocity is expressed as – \(\dfrac{\text { Distance }}{\text { Time }}\)
The UNITS for AREA should always be in SQUARE FEET and the UNITS for VELOCITY should always be in FEET PER SECOND.
Therefore, the equation should look like this
\[\dfrac{\text { cubic }}{\text { feet }}=\dfrac{\mathrm{ft}^{2}}{1} \times \dfrac{\mathrm{ft}}{\text { second }} \quad \rightarrow \quad \dfrac{\mathrm{ft} \times \mathrm{ft} \times \mathrm{ft}}{\text { second }}=\dfrac{\mathrm{ft} \times \mathrm{ft}}{1} \times \dfrac{\mathrm{ft}}{\text { second }} \nonumber \]
If you like using the “Pie Wheel” the following can be applied for Flow Rate.
Exercise 7.3
- What is the pressure in pounds per square inch at the bottom of a 45 foot tall water storage tank?
- An 80 foot tall standpipe is ¾ full. What is the pressure in psi at the bottom?
- A 30 foot water tank has a pressure gauge installed 2 feet from the bottom of the tank. If the pressure gauge reads 11 psi, what is the level in the tank?
- A 12” pipeline is flowing water at a velocity of 3.5 fps. What is the corresponding flow rate?
- A 36” diameter pipe is flowing water at a velocity of 5 fps. How many gallons of water will flow through it in 2 hours?
- An 18” pipe is flowing 1,500 gpm. What is the velocity?
- What is the area of a box culvert that is flowing 4.5 MGD at a velocity of 2.45 fps?
- An 8” diameter pipe is flowing water at a velocity of 7.65 fps. What is the flow rate in gallons per minute?
- A 28 foot tall water tank is sitting on a 75 foot tall hill. What is the pressure in a home at the bottom of the hill? (Assume the pressure gauge is 3 feet from the bottom of the home and the tank is full)
- What is the velocity through a 20” diameter pipe that flows 2.5 MGD?
- A home that is receiving water from a water tank that is on top of a 115 foot hill has a pressure of 57 psi. What is the water level in the tank?
- What is the area of a pipe that flows 2,550 gpm at a velocity of 5 fps?


<IP_ADDRESS>: What Copyright Covers
What Copyright Covers
Definition: Original Work of Authorship
Definition: Tangible Medium of Expression
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The kinds of works covered by copyright are listed in Section 102 of the Copyright Act. In order for a work to be covered by copyright, it must be an original work of authorship fixed in a tangible medium of expression. (See the detailed explanations below.)
There are several types of works that can be protected, including:
In addition to these, new formats such as email, software, video games, and digital works including web pages and online images have all been determined to have copyright protection.
Definition: Original Work of Authorship
In copyright law, originality means that a work is independently created and possesses at least a minimum amount of creativity. For example, an alphabetized list of names and phone numbers would not receive copyright protection because it required no creativity to produce.
Definition: Tangible Medium of Expression
For a work to be “fixed in a tangible medium,” it must exist in some perceptible format for more than a transitory duration. For example, a work that is fixed in a tangible medium could be written on paper, saved to a computer hard drive, or recorded on film. An improvised jazz performance that is not recorded would not have copyright protection, because the creative expression of the musician has not been saved in any tangible format.
Not all works are covered by copyright. Those not covered include:
Works already in the public domain (discussed in detail later in this book)
Works not fixed in a tangible medium
Ideas
Facts
Works of the U.S. government produced by government employees
If you create something as part of your job duties, it is likely a work made for hire. In these cases, the employer is considered the author and rights holder of a work made for hire rather than the employee.
Read the United States Copyright Office’s Works Made for Hire circular for a more nuanced discussion.


16.5F: Clot Retraction and Repair
Clot retraction is the shrinking of a blood clot facilitated by thrombolytic agents.
- Outline the process of clot retraction and repair
Key Points
- Clot retraction is dependent on the release of multiple coagulation factors, specifically Factor XIIIa at the end of the coagulation cascade.
- The formation of blood clots can cause a number of serious diseases. By breaking down the clot, the disease process can be arrested or the complications reduced.
- Clot retraction is the “shrinking” of a blood clot over a number of days. The edges of the blood vessel wall at the point of injury are slowly brought together to repair the damage.
- Clot retraction occurs due to the contraction, knotting, and twisting of the fibrin mesh.
- The steps of wound healing that follow clot retraction include inflammation, tissue proliferation, collagen and granulation tissue deposition, angiogenesis, wound contraction, and epithelialization.
Key Terms
- Clot retraction : The shrinking of a blood clot over the day following initial clot formation.
- angiogenesis : Growth of new blood vessels during wound healing.
- thrombus : Ablood clot formed from platelets and other elements that forms in a blood vessel in a living organism. It may cause thrombosis or obstruction of the vessel at its point of formation or travel to other areas of the body
The blood clots produced in hemostasis are merely the first step in repair and healing that occur after injury. Following a clot, inflammation draws leukocytes to the injury site to eliminate any pathogens that may have entered the body during the initial injury. Then, over the course of the next 24 hours, the clot retracts as tissue healing begins.
Clot Retraction
Thrombus or Blood Clot : Micrograph showing a thrombus (center of image) within a blood vessel of the placenta.
As the healing process occurs following blood clot formation, the clot must be destroyed in order to prevent thromboembolic events, in which clots break off from the endothelium and cause ischemic damage elsewhere in the body. By reducing the size of and breaking down the clot, the disease process can be arrested or the complications reduced.
Clot retraction refers to a regression in size of the blood clot over a number of days. During this process, the edges of the endothelium at the point of injury are slowly brought together again to repair the damage. Clot retraction is dependent on the release of multiple coagulation factors released at the end of the coagulation cascade, most notably factor XIIIa crosslinks. These factors cause the fibrin mesh to contract by forming twists and knots that condense the size of the clot. Clot retraction generally occurs within 24 hours of initial clot formation and decreases the size of the clot by 90%. Following clot retraction, a separate process called fibrinolysis occurs which degrades the fibrin of the clot while macrophages consume the expended platelets, thus preventing possible thromboembolism.
Wound Healing
While the clot retracts, the wound begins to heal. The first step of wound healing is epithelial cell migration, which forms a scab before the clot retracts. This occurs due to the stimulus of platelet-derived growth factor (PDGF). After clot retraction, true repair begins as tissue proliferation starts and collagen from the extracellular matrix is deposited in the wound while granulation tissue forms. Then new blood vessels grow into the healing tissue in a process called angiogenesis, which is stimulated by vascular endothelial growth factor (VEGF). The wound itself contracts, reducing in size. After these steps occur, new epithelial cells grow to cover the wound. If the wound was severe or unevenly shaped, or if healing takes too long, scarring may occur from collagen deposition. Most scarring on the skin is benign, but scarring inside the tissues of organs such as the heart or the lungs can cause health problems.


8.3: Cooperation Between States
In order to provide shared military and economic security as a unified entity, states engage in alliances. Military alliances help protect states from common enemies, and economic alliances allow for the free exchange of goods in a larger market. These alliances are also referred to as supranational organizations , and they all involve states giving up some of their sovereign power for the common good.
The largest supranational organization in the world is the United Nations (UN). Formed originally as the League of Nations after World War II, the UN now includes 193 states. The work of the UN includes peacekeeping, humanitarian relief, and the establishment of internationally approved standards of behavior. The headquarters of the UN is in New York City, and important subsidiary organizations of the UN include the World Health Organization (WHO), UNESCO (United Nations Educational, Cultural and Cultural Organization) and the Food and Agriculture Organization (FAO).
8.3.1 Military Alliances - NATO and Warsaw Pact
In terms of military alliances on a regional scale, the North Atlantic Treaty Organization (NATO) comprises 28 states and was developed after World War II to counter the threat of the former Soviet Union. Member states include numerous Western European states as well as the United States and Canada ( Figure 8.2. ).
When the Soviet Union existed, the Warsaw Pact was a military alliance between the Soviet Union and seven satellite states of Eastern and Central Europe ( Figure 8.2 ). The Warsaw Pact disbanded in 1991, and several of the former Soviet states as well as satellite states have subsequently joined NATO. As a result, Russia has felt isolated and vulnerable, and as a result, has been aggressively acting to seize or control territories in states close to its borders.
For example, in 2008, Russia engaged in a military conflict in Georgia, one of the former Soviet states, in order to support a separatist movement allied with Russia. In 2014, Russia invaded the peninsula of Crimea, within the territorial boundaries of Ukraine, one of the former states in the Soviet Union. Furthermore, Russia has intervened militarily against the rebel forces fighting in eastern Ukraine. In response to these provocations, NATO commenced Operation Atlantic Resolve, an ongoing series of training exercises between the United States and other NATO countries in former Warsaw Pact countries such as Poland, Romania, and Latvia.
8.3.2 Military Alliances – Terrorism
Although terrorists are characterized as non-military, non-state actors, they have a tremendous impact on states around the world and involve allied groups in many countries. Terrorism is the intimidation of a population by violence in order to further political aims. The first terrorist group with global influence is Al Qaeda, formed in 1988 by Osama bin Laden. Although its influence has waned in the past decade, Al Qaeda was responsible for the 9/11 attacks and had several affiliates including Boko Haram in Nigeria and Abu Sayyaf in the Philippines. With the rise of the Islamic State (ISIS/L) in Iraq and Syria in 2013, terrorist groups such as Boko Haram and Abu Sayyaf have consequently declared allegiance to ISIS/L. ISIS/L is an extremist Muslim group that intends to seize as much territory as possible in the Middle East and force their subjects to adhere to their strict version of Islamic fundamentalism. To facilitate their goal, ISIS/L conducts an extensive campaign on social media to recruit fighters to come to Syria and also conduct individual attacks in their home country. Their media campaign has been successful in recruiting numerous militants to come to Syria and to inspire or instigate attacks in the United States, France, Belgium, England, Sweden, Turkey, Afghanistan, Yemen, and Saudi Arabia ( Figure 8.3 ). Furthermore, ISIS/L isconducting regular military operations in Libya and Egypt.
To combat the Islamic State and other terrorist groups, interesting military alliances have developed, and the situation is exceptionally complex. The United States cooperates with numerous NATO allies to train and equip local military forces in Iraq and Syria. Iran, in alliance with Iraq, has provided assistance in driving ISIS/L out of Iraq. Kurdish forces in northern Iraq and Syria, in concert with NATO forces, have conducted much of the military action, and they hope for an independent state because of these efforts. Complicating this situation is the objection of the central governments of Iraq, Syria, and Turkey to an independent Kurdish state. Further clouding the situation is the alliance between Russia and the Syrian central government. Although these two countries do fight against ISIS/L, they are also opposed to other Syrian rebel forces, which the United States supports. The problem of ISIS/L propaganda is being addressed by numerous governments and social media outlets such as Facebook, Twitter, and YouTube. More importantly, though, military success on the ground against ISIS/L forces in Iraq and Syria is decreasing the influx of foreign fighters as well as the proliferation of internet propaganda.
8.3.3 Economic Alliances – European Union and NAFTA
One of the most prominent economic alliances in the world is the European Union (EU), which consists of 28 member states ( Figure 8.4 ). What began as the European Community (EC) in 1958, the European Union has grown significantly from the original six members and now includes seven Eastern European states that were formerly in the Soviet Union. The EU has developed a common currency, the euro, for all member countries and a European Central Bank. Furthermore, at most boundaries, a passport is not required to enter another country.
One of the weaknesses of the EU is the need to subsidize poorer countries, creating financial difficulties for the more wealthy members. For example, Greece has experienced large debts that have required rescue funding from the EU. Another issue confronting the EU is whether or not to allow Turkey to join, as Greece has long-standing disputes with Turkey over territory in Cyprus, and the Turkish central government has been accused of anti-democratic practices. Perhaps most concerning for the EU is the imminent departure of the United Kingdom (UK) from the alliance in an action termed “Brexit.” In 2016, the UK voted by referendum to leave the EU, and is on schedule to formally break away in 2019. The UK’s decision to leave the EU is not solely related to economics, as not only is the EU an economic alliance, but agreements on social and political policies are involved as well. The majority of British citizens are generally against the subsidizing of poorer states and the increasing number of immigrants who use scare public resources, and they generally desire greater autonomy. With the exit of Britain from the EU, EU members are concerned that other states may follow suit.
An important economic alliance for the United States is the North American Free Trade Agreement (NAFTA). Established in 1992, this alliance integrates the United States, Mexico and Canada and facilitates the flow of goods and services across borders. The Trump administration has repeatedly criticized this agreement, as manufacturers have relocated production to Mexico which has resulted in the loss of manufacturing jobs in America. Whether or not the United States withdraws from NAFTA or simply renegotiates the agreement remains to be seen.
Another significant alliance that is being considered is the Trans-Pacific Partnership, an agreement between 11 countries that border the Pacific Ocean, and originally included the United States. The proposal was signed in February 2017, but the United States had already withdrawn from the agreement in January of the same year, making ratification virtually impossible. The goal of the agreement is to promote economic prosperity by lowering tariff barriers, but also promote environmental and labor protections as well as protect intellectual property. Critics say that it would result in the loss of U.S. jobs and weaken the sovereignty of the U.S.


5.2: Lagrange's Theorem
We’re finally ready to state Lagrange’s Theorem, which is named after the Italian born mathematician Joseph Louis Lagrange. It turns out that Lagrange did not actually prove the theorem that is named after him. The theorem was actually proved by Carl Friedrich Gauss in 1801.
Let \(G\) be a finite group and let \(H\leq G\) . Then \(|H|\) divides \(|G|\) .
This simple sounding theorem is extremely powerful. One consequence is that groups and subgroups have a fairly rigid structure. Suppose \(G\) is a finite group and let \(H\leq G\) . Since \(G\) is finite, there must be a finite number of distinct left cosets, say \(H, a_2H, \ldots, a_{n}H\) . Corollary 5.1.2 tells us that each of these cosets is the same size. In particular, Lagrange’s Theorem implies that for each \(i\in\{1,\ldots, n\}\) , \(|a_iH|=|G|/n\) , or equivalently \(n=|G|/|a_iH|\) . This is depicted in Figure \(\PageIndex{1}\), where each rectangle represents a coset and we’ve labeled a single coset representative in each case.
One important consequence of Lagrange’s Theorem is that it narrows down the possible sizes for subgroups.
Suppose \(G\) is a group of order 48. What are the possible orders for subgroups of \(G\) ?
Lagrange’s Theorem tells us what the possible orders of a subgroup are, but if \(k\) is a divisor of the order of a group, it does not guarantee that there is a subgroup of order \(k\) . It’s not too hard to show that the converse of Lagrange’s Theorem is true for cyclic groups. However, it’s not true, in general.
Provide an example of a finite group \(G\) such that \(|G|\) has a divisor \(k\) but \(G\) does not have a subgroup of order \(k\) .
Using Lagrange’s Theorem, we can quickly prove both of the following theorems.
Let \(G\) be a finite group and let \(a\in G\) . Then \(|a|\) divides \(|G|\) .
The next exercise shows us that the converse of Theorem \(\PageIndex{2}\) is not true.
Argue that \(S_4\) does not have any elements of order 8.
For every prime \(p\) , if \(G\) has order \(p\) , then \(G\cong \mathbb{Z}_p\) .
For every prime \(p\) , there is a unique group of order \(p\) up to isomorphism.
Lagrange’s Theorem motivates the following definition.
Let \(G\) be a group and let \(H\leq G\) . The index of \(H\) in \(G\) is the number of cosets (left or right) of \(H\) in \(G\) . Equivalently, if \(G\) is finite, then the index of \(H\) in \(G\) is equal to \(|G|/|H|\) . We denote the index via \([G:H]\) .
Let \(H=\langle (1,2)(3,4),(1,3)(2,4)\rangle\) .
- Find \([A_4:H]\) .
- Find \([S_4:H]\) .
Find \([\mathbb{Z}:4\mathbb{Z}]\) .