Question

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find

(i) $A\times (B\cap C)$

(ii) $(A\times B)\cap (A\times C)$

(iii) $A\times (B\cup C)$

(iv) $(A\times B)\cup (A\times C)$

Answer 1

(i) We have,

$A=\{1,2,3\},B=\{3,4\}andC=\{4,5,6\}$

$\therefore B\cap C=\{3,4\}\cap \{4,5,6\}=\left\{4\right\}$

$\therefore A\times (B\cap C)=\{1,2,3\}\times \left\{4\right\}$

$=\left\{\right(1,4),(2,4),(3,4\left)\right\}$

$\Rightarrow A\times (B\cap C)=\left\{\right(1,4),(2,4),(3,4\left)\right\}$

(ii) We have,

A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}

$\therefore A\times B=\{1,2,3\}\times \{3,4\}$

$=\left\{\right(1,3),(1,4),(2,3),(2,4),(3,3),(3,4\left)\right\}$

and

$A\times C=\{1,2,3\}\times \{4,5,6\}$

= {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)}

$(A\times B)\cap (A\times C)=\left\{\right(1,4),(2,4),(3,4\left)\right\}$

(iii) We have,

A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}

$\therefore B\cup C=\{3,4\}\cup \{4,5,6\}$

= {3, 4, 5, 6}

$A\times (B\cup C)=\{1,2,3\}\times \{3,4,5,6\}$

$=\left\{\right(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,3),(3,4),(3,5),(3,6\left)\right\}$

(iv) We have,

$A=\{1,2,3\},B=\{3,4\}andC=\{4,5,6\}$

$\therefore A\times B=\{1,2,3\}\times \{3,4\}$

and,

$A\times C=\{1,2,3\}\times \{4,5,6\}$

$=\left\{\right(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6\left)\right\}$

$\therefore (A\times B)\cup (A\times C)=\left\{\right(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,3),(3,4),(3,5),(3,6\left)\right\}$