Let p : If x is an integer and x2 is even, then x is even.Using the m

1.	Let p : If x is an integer and x2 is even, then x is even.Using the method of contrapositive, prove that p is true.
Answer» Let p: x is an integer and x² is even. q: x is even For contrapositive, ~p = x is an integer and x² is not even. ~q = x is not even. Now, the statement is: If x is an integer and x² is not even, then x is not even. Proof: Let x be an odd integer and x = 2n + 1 ⇒x² = (2n + 1)² = 4n² + 4n + 1 (odd integer) Thus, if x is an integer and x² is not even, then x is not even.

Let p : If x is an integer and x2 is even, then x is even.Using the method of contrapositive, prove that p is true.

Answer»

Let p: x is an integer and x² is even.

q: x is even

For contrapositive,

~p = x is an integer and x² is not even.

~q = x is not even.

Now, the statement is: If x is an integer and x² is not even, then x is not even.

Proof:

Let x be an odd integer and x = 2n + 1

⇒x² = (2n + 1)² = 4n² + 4n + 1 (odd integer)

Thus, if x is an integer and x² is not even, then x is not even.