Suppose 𝑈 is the power set of the set 𝑆={1,2,3,4,5,6}. For any 𝑇∈𝑈, let |𝑇| denote the number of elements in 𝑇 and 𝑇′ denote the complement of 𝑇. For any 𝑇,𝑅∈𝑈, let 𝑇∖𝑅 be the set of all elements in 𝑇 which are not in 𝑅. Which one of the following is true?
A. ∀𝑋∈𝑈 (|𝑋|=|𝑋′|)
B. ∃𝑋∈𝑈 ∃𝑌∈𝑈 (|𝑋|=5,|𝑌|=5 and 𝑋∩𝑌=∅)
C. ∀𝑋∈𝑈 ∀𝑌∈𝑈 (|𝑋|=2,|𝑌|=3 and 𝑋∖𝑌=∅)
D. ∀𝑋∈𝑈 ∀𝑌∈𝑈 (𝑋∖𝑌=𝑌′∖𝑋′)
Answer:D
Suppose 𝑈 is the power set of the set 𝑆={1,2,3,4,5,6}. For any 𝑇∈𝑈, let |𝑇| denote the number of elements in 𝑇 and 𝑇′ denote the complement of 𝑇. For any 𝑇,𝑅∈𝑈, let 𝑇∖𝑅 be the set of all elements in 𝑇 which are not in 𝑅. Which one of the following is true?
A. ∀𝑋∈𝑈 (|𝑋|=|𝑋′|) B. ∃𝑋∈𝑈 ∃𝑌∈𝑈 (|𝑋|=5,|𝑌|=5 and 𝑋∩𝑌=∅) C. ∀𝑋∈𝑈 ∀𝑌∈𝑈 (|𝑋|=2,|𝑌|=3 and 𝑋∖𝑌=∅) D. ∀𝑋∈𝑈 ∀𝑌∈𝑈 (𝑋∖𝑌=𝑌′∖𝑋′)