Subjective Type

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) $$(A \cap B)' = A' \cup B'$$

Solution

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

(i) $$(A\cup B)'=\left \{2,3,4,5,6,7,8\right \}'=\left \{1,9\right \}$$
$$A'\cap B'=\left \{1,3,5,7,9\right \}\cap \{1,4,6,8,9\}=\left \{1,9\right \}$$
$$\therefore (A\cup B)'=A'\cap B'$$

(ii) $$(A\cap B)'=\left \{2\right \}'=\left \{1,3,4,5,6,7,8,9\right \}$$
$$A'\cup B'=\left \{1,3,5,7,9\right \}\cup \left \{1,4,6,8,9\right \}=\left \{1,3,4,5,6,7,8,9\right \}$$
$$\therefore (A\cap B)'=A'\cup B'$$

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