# Thread: Basic Question on Negation, Inverse and Contrapositive

1. ## Basic Question on Negation, Inverse and Contrapositive

Hello to all.

In a routine exercise I have been given the following statement:

Let k be an integer, if k^2 is a multiple of 2, then k is a multiple of 2.

I am then asked to determine the negative, converse, inverse and contrapositive of the aforementioned statement. The difficulty or doubt I am having is with the first portion of the statement (Let k be an integer...), specifically how or if it varies.

The following are the rules I used to determine my answers:
Starting with A=>B,

Negation: not (A=>B) is log.equiv. to (A and not B)
Converse: B=>A
Inverse: (not A) => (not B)
Contrapositive: (not B) => (not A)

With that in mind, the following are my findings:

Negation:
Let k be an integer, if k^2 is a multiple of 2 and k is not a multiple of 2.

Converse:
Let k be an integer, if k is a multiple of 2, then k^2 is a multiple of 2.

Inverse:
Let k be an integer, if k^2 is not a multiple of 2 then k is not a multiple of 2.

I wish to thank everyone in advance who offers help in determining if I am on the right path. Good day to all.
Contrapositive:
Let k be an integer, if k is not a multiple of 2 then k^2 is not a multiple of 2.

2. Originally Posted by gate13
In a routine exercise I have been given the following statement:
Let k be an integer, if k^2 is a multiple of 2, then k is a multiple of 2.
I am then asked to determine the negative, converse, inverse and contrapositive of the aforementioned statement.
Negation:
Let k be an integer, if k^2 is a multiple of 2 and k is not a multiple of 2.
You have one mistake. There is no if in the negation.

The negation: There is an integer $J$ such that $J^2$ is a multiple of 2 and $J$ is not a multiple of 2.

3. Thank you Plato for your quick response. Have a good day.