I am having trouble with the wording of these statements particularly the negation statement. Is that the best way to put it or could you provide a better alternative?

Also for the converse proof can you verify I disproved it? Thanks in advance; I'm new to proofs and any feedback is greatly appreciated.

Write the contrapositive to the statement: If $a,b,c\in \mathbb{N}$ and $a+b+c$ is even then $abc$ is even.

If $abc$ is odd then $a,b,c\notin \mathbb{N}$ or $a+b+c$ is odd.

Write the negation to the statement: For every real number $x$, there exists a real number $y$ such that $xy=1$.

For every real number $x$, and for every real number $y$ such that $xy\neq 1$

Write the converse to the statement and disprove it: If $a$ and $b$ are even integers then $a+b$ is even.

If $a+b$ is even then $a$ and $b$ are even integers.

By counter example:

Suppose $a,b\in \mathbb{Z}$ such that $a+b= 2$ its possible that $a=-1$ and $b=3$.

So if a+b is even there exists $a,b\notin 2\mathbb{Z}$.


The negation should be: there exists a real number $x$ for all $y$ such that $xy\neq 0$.


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