18. Five marks exercise problems IX

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.]