Question

If $U=\{a,b,c,d,e,f,g,h\},$
$P=\{a,b,c,f\},Q=\{d,g,f,h\},$
$\text{then which of the following are correct?}$

• A. $(P\cup Q{)}^{\prime }=\{e\}$
• B. $(P-Q{)}^{\prime }=\{d,e,f,g,h\}$
• C. $(P\cup Q{)}^{\prime }=P^{\prime }\cap Q^{\prime }$
• D. $P\cap {\varphi }^{\prime }=\{a,b,c,f\}$

Answer 1

Answer: (A), (B), (C), (D)

$P\cap {\varphi }^{\prime }=\{a,b,c,f\}$


From the above venn diagrams we can see that $(P\cup Q{)}^{\prime }=\{e\}.$

By drawing the venn diagrams we can see that $(P-Q{)}^{\prime }=\{d,e,f,g,h\}.$

$(P\cup Q{)}^{\prime }=P^{\prime }\cap Q^{\prime }$ is the DeMorgan's law and if needed can be verified using venn diagram.

${\varphi }^{\prime }=U,\text{s}oP\cap {\varphi }^{\prime }=P\cap U=P=\{a,b,c,f\}.$