21. Five marks exercise problems X

Let $A =$ The set of all natural numbers less than $8$, $B =$ The set of all prime numbers less than $8$, $C =$ The set of even prime numbers. Verify that $(A \cap B) \times C = (A \times C) \cap (B \times C)$

 

Answer:

 

To prove:

 

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

 

Explanation:

 

$A \cap B =$ $\left\{i,i,i,i\right\}$

 

$(A \cap B) \times C = \{$ $\}$

 

$A \times C = \{$ $\}$

 

$B \times C =\{$ $\}$

 

$(A \times C) \cap (B \times C) = \{$ \}\)

 

As a result, $(A \cap B) \times C = (A \times C) \cap (B \times C)$

 

Hence, we proved.

 

[Note: Enter the first and the second coordinates of the ordered pairs in the increasing order.]