Suppose π‘ˆ is the power set of the set 𝑆 = {1,2,3,4,5,6}. For any 𝑇 ∈ π‘ˆ,…

GATE Β· 2015 Β· CS Β· Set 3 Β· Computer Science & IT

Suppose π‘ˆ is the power set of the set 𝑆 = {1,2,3,4,5,6}. For any 𝑇 ∈ π‘ˆ, let |𝑇| denote the number of elements in 𝑇 and 𝑇′ denote the complement of 𝑇. For any 𝑇, 𝑅 ∈ π‘ˆ, let 𝑇 βˆ– 𝑅 be the set of all elements in 𝑇 which are not in 𝑅. Which one of the following is true?

  1. A.

    βˆ€π‘‹ ∈ π‘ˆ (|𝑋| = |𝑋′ |)

  2. B.

    βˆƒπ‘‹ ∈ π‘ˆ βˆƒπ‘Œ ∈ π‘ˆ (|𝑋| = 5, |π‘Œ| = 5 and 𝑋 ∩ π‘Œ = βˆ…)

  3. C.

    βˆ€π‘‹ ∈ π‘ˆ βˆ€π‘Œ ∈ π‘ˆ (|𝑋| = 2, |π‘Œ| = 3 and 𝑋 βˆ– π‘Œ = βˆ…)

  4. D.

    βˆ€π‘‹ ∈ π‘ˆ βˆ€π‘Œ ∈ π‘ˆ (𝑋 βˆ– π‘Œ = π‘Œβ€² βˆ– 𝑋′ )

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