Translating $\forall$ and $\exists$ in a statement The following statement has two versions – one where $d$ is quantified by $\forall$ and the second where it's quantified by $\exists$. The task here is to find a counterexample  where the statements below are false. The domain is all integers.


*

*$\forall a \forall b \forall c \forall d(a^d + b^d = c^d$)

*$\forall a \forall b \forall c \exists d(a^d + b^d = c^d$)


The first statement is false for at least some values of the variables. When $a=1, b=2, c=3, d=4$, the statement does not hold for all variables $a$, $b$, $c$, and $d$. For instance: 
$$
1^4 + 2^4 \ne 3^4
$$
The second statement is false when $a = 1, b = 2, c = 10$ because there doesn't exist a $d$ where the statement would be true. For instance, if $d$ was $5$:
$$
1^5 + 2^5 \ne 10^5
$$
Am I interpreting this correctly and do my counterxamples make sense?

The counterexample you gave for the first statement is correct.
To prove the second statement is false, you need to prove that its negation is true.  This negation is the statement $\exists a, \exists b, \exists c, \forall d, a^d+b^d \ne c^d$.  Take $a=b=0$, $c=1$.  Observe that for all integers $d$, $0^d + 0^d \ne 1^d$.  
