Question

Find a counterexample to show that the following statement is false. For all sets A and B, (A ∪ B)c = Ac ∪ Bc. Assume that all sets are subsets of a universal set U = {1, 2, 3, 4, 5}. (Enter your answer as a comma-separated list in the form A, B where both A and B are written in set-roster notation. Enter EMPTY or ∅ for the empty set.)

Answer 1

Answer:

Given,

The universal set,

U = {1, 2, 3, 4, 5},

Let A = {1, 2},

B = {2, 3, 4}

$A\cup B=\text{ all elements of A and B }=\{1, 2, 3, 4\}$

$(A\cup B)^c=U-(A\cup B) = \{1, 2, 3, 4, 5\} - \{1, 2, 3, 4\} = \{5\}$

Now,

$A^c = U - A = \{3, 4, 5\}$

$B^c=U-B=\{1, 5\}$

$A^c\cup B^c = \{1, 3, 4, 5\}$

$\implies (A\cup B)^c\neq A^c\cup B^c$

Hence, proved......