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?