Question

If $A=\{1,2,3\},B=\left\{4\right\},C=\left\{5\right\},$

then verify that:

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

Answer 1

Given : $A=\{1,2,3\},B=\left\{4\right\},C=\left\{5\right\},$

Now,

$B\cup C=\left\{4\right\}\cup \left\{5\right\}$

$\Rightarrow (B\cup C=\{4,5\}$

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

$\Rightarrow A\times (B\cup C)=\left\{\begin{array}{cc}(1,4)(1,5),(2,4),(2,5),& \\ (3,4),(3,5)& \end{array}\right\}$

Also,

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

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

$A\times C=\{1,2,3\}\times \left\{5\right\}$

$\Rightarrow A\times C=\left\{\right(1,5),(2,5),(3,5\left)\right\}$

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

$\cup \left\{\right(1,5),(2,5),(3,5\left)\right\}$

$\Rightarrow (A\times B)\cup (A\times C)=\left\{\begin{array}{cc}(1,4),(2,4),(3,4),(1,5),& \\ (2,5),(3,5)& \end{array}\right\}$

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

Hence verfied