Subjective Type

If $$A = \{ 3, 5, 7, 9, 11 \}, B = \{7, 9, 11, 13\}, C = \{11, 13, 15\}$$ and $$D = \{15, 17\}; $$find (i) $$A \cap B$$ (ii) $$B \cap C$$ (iii) $$A \cap C \cap D$$ (iv) $$A \cap C$$ (v) $$B \cap D$$ (vi) $$A \cap (B \cup C)$$ (vii) $$A \cap D$$ (viii) $$A \cap (B \cup D)$$ (ix) $$( A \cap B ) \cap ( B \cap C )$$ (x) $$(A \cup D) \cap ( B \cup C)$$

Solution

$$A = \{ 3, 5, 7, 9, 11 \}, B = \{7, 9, 11, 13\}, C = \{11, 13, 15 \}$$ and $$D = \{15, 17\}$$

(i) $$A\cap B=\left \{7,9,11\right \}$$

(ii) $$B\cap C=\left \{11,13\right \}$$

(iii) $$A\cap C\cap D=(A\cap C ) \cap D=\left \{11\right \}\cap \left \{15, 17\right \}=\phi$$

(iv) $$A\cap C=\left \{11\right \}$$

(v) $$B\cap D=\phi$$

(vi) $$A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$$
$$=\left \{7,9,11\right \}\cup \left \{11\right \}=\left \{7,9,11\right \}$$

(vii) $$A\cap D=\phi$$

(viii) $$A\cap (B\cup D)=(A\cap B)\cup (A\cap D)$$
$$=\left \{7,9,11\right \}\cup \phi =\left \{7,9,11\right \}$$

(ix) $$(A\cap B)\cap (B\cup C)=\left \{7,9,11\right \}\cap \left \{7,9,11,13,15\right \}=\left \{7,9,11\right \}$$

(x) $$(A\cup D)\cap (B\cup C)=\left \{3,5,7,9,11,15,17\right \}\cap \left \{7,9,11,13,15\right \}$$
$$=\left \{7,9,11,15\right \}$$

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