Q:

Mathematical notation and contrapositives

(a) Write the statement
“No integer bigger than 70 can be written as the sum of
3 integers smaller than 30”
using only mathematical symbols (you may need to use quantifiers). Is the
statement True or False? You should carefully justify your answer.
(b) Write the contrapositive of the statement in (a), again using only mathematical
symbols. Is this statement true or false? Why?
How are you supposed to write the contrapositive if there is no implication in the statement?
Question is asked in more detail whilst seeking more answers on more concepts

A:

In order to write your answer comprehensively with mathematical symbols added in,  we must atomize your statement.
$\forall$ means "for all", $\exists$ means "there exists", $\wedge$ means "and". There is no mathematical symbol that I know for "such that".
"No integer greater than 70" $\Longrightarrow x \in \mathbb{Z}\ \wedge\ x>70$
"x can be written as the sum of three integers smaller than 30" $\Longrightarrow \exists a,b,c < 30 $ such that $ a+b+c=x$.
"x cannot be written as the sum of three integers smaller than 30" $\Longrightarrow \forall a,b,c < 30 , a+b+c \neq x$.
The final statement, therefore, is $x \in \mathbb{Z}\ \wedge\ x>70 \implies \forall a,b,c < 30 , a+b+c \neq x$.
Of course, this statement is not true: Take $x=71,a=22,b=23,c=26$.
Now, we go to the negation : How do we negate this statement? Let's again do it piece by piece. First of all, because the statement is false, it's negation must be true. (this is sometimes called the law of the excluded middle, but every statement is either true or false,isn't it?)
Now, to negate, how did we find a counterexample to the above statement? We found a number $x$, greater than 70, and found a triple of numbers $(a,b,c)$ less than 30, such that $x=a+b+c$. Now, we know the negation : There is a number $x>70$ , and a set of numbers $a,b,c <30$, such that $x=a+b+c$. We write this as $\exists x > 70,x \in \mathbb{Z}, \wedge \exists a,b,c < 30$ such that $x=a+b+c$. Of course, the negation is true, as we showed with $x=71$.
The contrapositive of this statement is also simple: the contrapositive of $p \implies q$ is $\neg q \implies \neg p$. Now you see what to do: negate the earlier two mini-statements, and reverse the implication! That is, 
Negation of "No integer greater than 70" $\Longrightarrow x \in \mathbb{Z}\ \wedge\ x<70$
Negation of "x can be written as the sum of three integers smaller than 30" $\Longrightarrow \forall a,b,c < 30 , a+b+c \neq x$.
The contrapositive: $\forall a,b,c < 30 , a+b+c \neq x \implies x<70$ (forget about $x \in \mathbb{Z}$,  it's not important, and will follow all the statements like a blind man).
Is the contrapositive false? Of course it is! Again, take our $71$ example. $22,23$ and $26$ were three numbers smaller than $30$, but their sum was greater than $70$, namely $71$! So the contrapositive is also false!
I hope you now have a good idea of how to negate statements. Hope this has helped you. Please reply back if any doubts.

