Question

Let U= {1, 2, 3, 4,5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B= {2, 4, 6, 8} and C={3, 4, 5, 6} Find $(A \cup C)^{'}$

Answer 1

A U C = {1,2,3,4} U {3,4,5,6}

         ={1,2,3,4,5,6}

(A U C)' = U -(A U C)

          ={1,2,3,4,5,6,7,8,9} - {1,2,3,4,5,6}

          ={7,8,9}

hope its helpfull

Answer 2

concept : A' is known as absolute complement of A. means, A' is the set of all the objects or elements that do not belong to A. in figure, shaded region indicates A' or absolute complement of A.
for example :- if A = {a, b, c , d} and U = {a, b, c, d, e, f, ....z} Then, A' = U - A = {e, f, g, .... z}.

Here, given, U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {1, 2, 3, 4} and C = {3, 4, 5, 6}
first of all, we have to find $A\cup C$
$A\cup C$ = {1, 2, 3, 4} U {3, 4, 5, 6}
= {1, 2, 3, 4, 5, 6}

now, $(A\cup C)'=U-(A\cup C)$
= {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4, 5, 6}
= {7, 8, 9}