# Math Help - Negation clarification

1. ## Negation clarification

x is a positive odd integer --> x + 2 is a positive odd integer

The trouble comes in negating each statement when finding the contrapositive

do i negate the statements to:

x is a NEGATIVE odd integer
x + 2 is a NEGATIVE odd integer

or

x is a positive EVEN integer
x + 2 is a positive EVEN integer

or

x is a NEGATIVE EVEN integer
x + 2 is a NEGATIVE EVEN integer

or (likely not but i just thought i'd ask) U = universe
x is a positive odd U-Z
x + 2 is a positive odd U-Z

I know that a statement has a truth value, but my prof and the text never went over what parts of a statement are eligible for negation. Is it right to say that the statements I have used as examples are in fact statements even though x is unknown, because they are quantified?

Am I correct to say that the quantification of these statements is implicit rather than explicit?

Argh this seems like a lot of questions, I'd really appreciate it someone could answer at least a few of them if answering them all is too laborious.

Thanks

2. Originally Posted by Noxide
x is a positive odd integer --> x + 2 is a positive odd integer

The trouble comes in negating each statement when finding the contrapositive
Negating a statement has nothing to do with positive or negative numbers. If $A \Rightarrow B$ then the contrapositive is $\text{not }B \Rightarrow \text{ not } A$.

So here, "if $x$ is a positive odd integer then $x + 2$ is a positive odd integer" is negated by putting "not" before the condition in each statement and switching them. That is, "if $x+2$ is not a positive odd integer then $x$ is not a positive odd integer".

3. Thanks.

Can you tell me a little bit more about the condition of a statement. ie How it can be found more easily.

Idk why my text doesn't go over this.