Single Choice

Let $$A$$ and $$B$$ be two sets then $$\left( A\cup B \right) '\cup \left( A'\cap B \right) $$ is equal to

A$$A'$$
Correct Answer
B$$A$$
C$$B'$$
D$$A\cap A'$$

Solution

Let $$S=(A\cup B)'\cup (A'\cap B)$$
According to De Morgan' Law, we have
$$S=(A'\cap B')\cup (A'\cap B)$$
$$\Rightarrow S=(A'\cup A')\cap (A'\cup B)\cap (B'\cup A')\cap (B'\cup B)$$
$$\Rightarrow S=A'\cap \left \{ A'\cup (B\cap B') \right \}\cap \bigcup $$
$$\Rightarrow S=A'\cap (A'\cup \phi )\cap \bigcup $$
$$S=A'\cap (A'\cup \phi )\cap \bigcup $$
$$\therefore S=A'\cap A'\cap \bigcup =A'\cap \bigcup =A'$$

SIMILAR QUESTIONS

Sets, Relations and Functions

If $$X$$ and $$Y$$ are two sets, $$X\cap { \left( Y\cup X \right) }^{ C }$$ is equal to

Sets, Relations and Functions

The set $$\left( A\cap { B }^{ C } \right) ^{ C }\cup \left( B\cap C \right) $$ is equal to

Sets, Relations and Functions

If $$U=\{1,2,3,4,5,6, 7, 8, 9 \}, A = \{2, 4, 6, 8\} $$ and $$B = \{ 2, 3, 5, 7\}.$$ Verify that (i) $$(A \cup B)' = A' \cap B'$$ (ii) $$(A \cap B)' = A' \cup B'$$

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