Subjective Type

If $$U=\left\{a,b,c,d,e \right\}, A=\left\{a,b,c \right\}$$ and $$B=\left\{b,c,d,e \right\}$$ the verify that : $$(A\cup B)' =(A' \cap B')$$

Solution

$$A\cup B$$ is the set of elements which are in $$A$$ or $$B$$ or in both
$$A\cap B$$ is the set of all elements common to both $$A$$ and $$B$$
$$A'$$ is the set of all elements which are not in $$A$$
$$B'$$ is the set of all elements which are not in $$B$$
$$U=\{a,b,c,d,e\},\,A=\{a,b,c\}$$ and $$B=\{b,c,d,e\}$$
Now,
$$A\cup B=\{a,b,c,d,e\}$$
$$(A\cup B)'=\phi$$ --- ( 1 )
$$A'=\{d,e\}$$
$$B'=\{a,b\}$$
$$(A'\cap B')=\phi$$ --- ( 2 )
From ( 1 ) and ( 2 ),
$$\Rightarrow$$ $$(A\cup B)'=(A'\cap B')$$

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