Question

If $X=\{1,2,3,4,5,6,7,8,9,10\}$ is the universal set and$A=\{1,2,3,4\},B=\{2,4,6,8\},C=\{3,4,5,6\}$ verify the following.
(a) $A\cup (B\cup C)=(A\cup B)\cup C$
(b)$A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$
(c) $(A^{\prime }{)}^{\prime }=A$

• A. Only a is true
• B. Only b and c are true
• C. Only a and b are true
• D. All three a,b and c are true.

Answer 1

The correct option is D All three a,b and c are true.

given

$X=${$1,2,3,4,5,6,7,8,9,10$},$A=\{1,2,3,4\},B=\{2,4,6,8\},C=\{3,4,5,6\}$

now from the given data

$A\cup B=$$\{1,2,3,4,6,8\}$,$B\cup C=$$\{2,3,4,5,6,8\}$

Now $A\cup (B\cup C)=\{1,2,3,4,5,6,8\}and(A\cup B)\cup C=$$\{1,2,3,4,5,6,8\}$

$\therefore A\cup (B\cup C)=(A\cup B)\cup C=\{1,2,3,4,5,6,8\}$

$A\cap B=$$\{2,4\}$,$A\cap C=$$\{3,4\}$,$B\cup C=$$\{2,3,4,5,6,8\}$

Now $A\cap (B\cup C)=\{2,3,4\}and(A\cap B)\cup (A\cap C)=$$\{2,3,4\}$

$\therefore A\cap (B\cup C)=(A\cap B)\cup (A\cap C)=$$\{2,3,4\}$

$A^{\prime }=$$\{5,6,7,8,9,10\}$

therefore $(A^{\prime }{)}^{\prime }=$$\{1,2,3,4\}$$=A$

therefore all three a,b,c are true