Set, Relations and FunctionsPYQ Nov. 19Question 1889 of 217
All Questions

(Ac)c=?\displaystyle (A^c)^c = ? Note: This que is from matrix (deleted topic).

Options

AA\displaystyle A
BAc\displaystyle A^c
CAT\displaystyle A^T
DA2T\displaystyle A^{2T}
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Correct Answer

✅ Option a — A\displaystyle A

All Options:

  • AA\displaystyle A
  • BAc\displaystyle A^c
  • CAT\displaystyle A^T
  • DA2T\displaystyle A^{2T}

Detailed Solution & Explanation

In set theory, let U\displaystyle U be the universal set and A\displaystyle A be any subset of U\displaystyle U.
The complement of set A\displaystyle A, denoted by Ac\displaystyle A^c, is the set of all elements in U\displaystyle U that do not belong to A\displaystyle A:
Ac={x∈U∣x∉A}A^c = \{x \in U \mid x \notin A\}
Taking the complement of Ac\displaystyle A^c again:
(Ac)c={x∈U∣x∉Ac}(A^c)^c = \{x \in U \mid x \notin A^c\}
Since x∉Ac\displaystyle x \notin A^c implies that x∈A\displaystyle x \in A, we have:
(Ac)c=A(A^c)^c = A
(Note: Even if interpreted as matrices where c\displaystyle c represents the transpose T\displaystyle T, the double transpose of a matrix A\displaystyle A returns the original matrix: (AT)T=A\displaystyle (A^T)^T = A.)
Hence, **Option A** is the correct answer.

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