Question

If A = {2, 3},B = {3, 4} and C = {4, 5}, show that A × (B ∩ C) = (A × B) ∩ (A × C).

Answer 1

$\large\underline{\sf{Given- }}$

A = {2, 3}

B = {3, 4}

and

C = {4, 5}

$\large\underline{\sf{To\:show - }}$

A × (B ∩ C) = (A × B) ∩ (A × C).

$\large\underline{\sf{Solution-}}$

Given that,

A = {2, 3}

B = {3, 4}

and

C = {4, 5}

Consider,

$\rm :\longmapsto\:B\cap C = \{3,4\}\cap \{4,5\} \: = \: \{4\}$

Now,

$\rm :\longmapsto\:A \times (B\cap C)$

$\rm \: = \:\{2,3\} \times \{4\}$

$\rm \: = \:\{(2,4), \: (3,4)\}$

So,

$\red{\rm :\longmapsto\:A \times (B\cap C) = \:\{(2,4), \: (3,4)\} - - - (1)}$

Now, Consider

$\rm :\longmapsto\:A \times B$

$\rm \: = \:\{2,3\} \times \{3,4\}$

$\rm \: = \:\{(2,3),(2,4),(3,3),(3,4)\}$

$\bf\implies \:A \times B= \:\{(2,3),(2,4),(3,3),(3,4)\}$

Now, Consider

$\rm :\longmapsto\: A\times C$

$\rm \: = \:\{2,3\} \times \{4,5\}$

$\rm \: = \:\{(2,4),(2,5),(3,4),(3,5)\}$

$\bf\implies \:A \times C = \:\{(2,4),(2,5),(3,4),(3,5)\}$

Hence,

$\bf\implies \:(A \times B)\cap (A \times C) = \:\{(2,4),(3,4)\} - - (2)$

So, from equation (1) and equation (2), we concluded that

$\bf\implies \:A \times (B\cap C) = (A \times B)\cap (A \times C)$

Additional Information :-

$\boxed{ \bf{ \: A \times B \: \ne \: B \times A}}$

$\boxed{ \bf{ \: n(A \times B) = n(A) \times n(B)}}$

$\boxed{ \bf{ \: A \times \phi \: = \: \phi}}$