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.}


→x2 = (2n+1)2


→x2 = 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.


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