Negating statements help I need to negate (move the negation inside) or write the contrapositive statements of the following statements:


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*Write the contrapositive of: If everyone is here, then someone will leave.

*Write the negation of: If Alice and Bob go, then Carol or Dave will come.

*Write the negation of Every integer is even or odd, but no integer is even and odd.

*rewrite as if then statement, then write its contrapositive: Every integer bigger than $1$ is divisible by some prime.

*rewrite as if then statement, then negate statement:  Every Integer that is divisible by $2$ and $3$ is divisible by $6$.
This is what I’ve done:


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*If someone didn’t leave, then not everyone was here. 

*Alice and Bob went; however, neither Carol or Dave came along.

*There is an integer that is neither even or odd, or there is an integer that is even or odd.

*If $x$ is an integer bigger than $1$, then it is divisble by some prime. Negation: 
$x$ is an integer bigger than $1$, however $x$ is not divisible by any prime. 

*If $x$ is an integer divisible by $2$ and $3$ then $x$ is also divisible by $6$. 
contrapositive: If $x$ is an integer not divisible by $6$, then $x$ is not divisble by $2$ and $3$.
These are not entirely correct, so I was hoping you guys would tell me what's wrong with them and how to fix them.

You need to break the statements into smaller pieces. I will give you an example with the first statement but you will have to do the rest by yourself:
If everyone is here, then someone will leave.
There are two statements here: "$A = \text{ everyone is here }$" and "$B = \text{ someone will leave }$". Then the statement you're given is "If $A$ then $B$" which means that whenever $A$ happens $B$ must happen. Therefore the negation would be that $A$ happens and $B$ does not happen. Formally the negation is "$A$ and not $B$".
Negating $B$ is easy $\text{not } B = \text{nobody will leave}$. Therefore $A$ and not $B$ means:
$\text{everyone is here and nobody will leave.}$

Note that I'm not a native English speaker. 
