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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.
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.
RS Agarwal Maths Class 11 Mathematical Reasoning
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.

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.
The contrapositive statement is: If x is an integer and x2

is not even, then x is not even.

Proof;
Let x be an odd/ not even integer
x = 2n + 1
{2n must be an even integer as when an integer is multiplied with an even integer, the
answer is always even. Adding one ensures that the integer is odd after the
multiplication.}
→x
2 = (2n+1)2
→x
2 = 4n2 + 4n + 1
{4n2 and 4n are even irrespective of integer n’s value. Adding 1 makes the not
even/odd}
Thus, if x is an integer and x2

is not even, then x is not even.

 

i

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