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.




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



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


"Did not" is not the complement for "will".  "Was not" is not the complement for "is".  Don't change the tenses.




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




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


Don't tamper with the verbs; just leave them as they are, or negate them if required.   "Go" either remains "go" or negates to "did not go". So on.




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




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


The negation of "all are" is "some are not", but the negation of "not some are" is just "some are".
Also the negation of "are even or odd" is "are not-even and not-odd", which you may say as "are odd and even".




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




*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.

Where did $x$ come from?  Don't introduce terms not used in the original statement.
Start with "If any integer is blah blah blah, then it is rhubarb rhubarb," then negate that.
Hint: That is still a universal quantified statement.




*rewrite as if then statement, then negate statement:  Every Integer that is divisible by $2$ and $3$ is divisible by $6$.




*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 $3$.

Same. Try again
