Set Notation

Sets - Problem Solving


Rakesh is classifying flags of the world according to the presence of red or blue in them.

Which flag would be in the same category as that of South Korea?

Given the sets U={1,2,3,,8},A={X8X,XU},B={Y1Y,YU}, \begin{aligned} U &=\{1,2,3,\ldots,8\},\\ A &=\{X \mid 8 \in X, X \subseteq U\},\\ B &=\{Y \mid 1 \in Y, Y \subseteq U\}, \end{aligned} what is AB?\lvert{A \cup B}\rvert?

Details and assumptions

S\lvert{S}\rvert denotes the number of members of the set S.S.

Let BB be a subset of the set A={1,2,3,4,5,,n}A=\{1,2,3,4,5,\ldots,n\} such that 11 and 22 are elements in BB but 33 and 44 are not. If there are a total of 128128 such subsets BB, what is the value of n?n?

In counting the numbers from 1 to 99 (inclusive), what is the total number of times that the digit 3 is used?

Details and assumptions
The number 1212 uses the digit 1 once and the digit 2 once.
The number 111111 uses the digit 1 three times.

Given the set U={xx is a positive integer 12},U=\{x \mid x \text{ is a positive integer } \leq 12\}, let A={2,4}A=\{2, 4\} and B={2,6,9,11,12}B=\{2, 6, 9, 11, 12\} be two subsets of U.U. How many sets XX are there such that I.XU,II.BX=3,III.A and X are mutually disjoint?\begin{aligned} \text{I}&. && X \subseteq U, \\ \text{II}&. && |B \cap X|=3, \\ \text{III}&. && A \text{ and } X \text{ are mutually disjoint?} \end{aligned}

Details and assumptions

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