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

(i) $A^{\prime }$ (ii) $B^{\prime }$ (iii) ${(A\cup C)}^{\prime }$

(iv) ${(A\cup B)}^{\prime }$ (v) $(A{)}^{\prime }$ (vi) ${(B-C)}^{\prime }$

Answer 1

Here 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}

(i) $A^{\prime }$ = U - A

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

= {5, 6, 7, 8, 9}

(ii) $B^{\prime }=U-B$

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

= {1, 3, 5, 7, 9}

(iii) $A\cup C$ = {1, 2, 3, 4} $\cup$ {3, 4, 5, 6}

= {1, 2, 3, 4, 5, 6}

${(A\cup C)}^{\prime }$ = U - $(A\cup C)$

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

= {7, 8, 9}

(iv) $A\cup B$ = {1, 2, 3, 4} $\cup$ {2, 4, 6, 8}

= {1, 2, 3, 4, 6, 8}

${(A\cup B)}^{\prime }=U-(A\cup B)$

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

= {5, 7, 9}

(v) $A^{\prime }=U-A$

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

= {5, 6, 7, 8, 9}

$(A^{\prime }{)}^{\prime }=U-A^{\prime }$

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

= {1, 2, 3, 4} = A.

(vi) B - C = {2, 4, 6, 8} - {3, 4, 5, 6} = {2, 8}

${(B-C)}^{\prime }=U-(B-C)$

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

= {1, 3, 4, 5, 6, 7, 9}