For the universal set $U = a, b, c, d, e$ , let $A = a, b$ and $B = c, d$ . Find $(A c u pB)^{c}$ and verify that it equals $A^{c} c a p B^{c}$ .

hard 2 marks Class 11 · Mathematics · Sets

Answer

Both

(A \cup B)^{c}

and

A^{c} \cap B^{c}

equal

{e}

The correct answer is Both $(A \cup B)^{c}$ and $A^{c} \cap B^{c}$ equal ${e}$ — Both sides are equal to ${e}$ , confirming De Morgan’s law.

Step-by-step solution

Compute $A \cup B = {a, b, c, d}$ .
Take its complement in $U$ : $U ∖ {a, b, c, d} = {e}$ . So $(A \cup B)^{c} = {e}$ .
Compute $A^{c} = {c, d, e}$ and $B^{c} = {a, b, e}$ .
Intersect them: ${c, d, e} \cap {a, b, e} = {e}$ . Thus $A^{c} \cap B^{c} = {e}$ .
Both sides are equal to ${e}$ , confirming De Morgan’s law.

Common mistake: taking complement relative to the wrong universal set or mis‑computing the intersection

Concept tested: De Morgan’s law for set complements