Given two non-trivial (not $\emptyset$ or $\Sigma^*$) languages $A$, $B$ over an alphabet $\Sigma$, which of the following is correct:
a. There is a language $C$ such that $A\leq_pC$ and $B\leq_pC$.
c. There is a language $C$ such that $C\leq_pA$ and $C\leq_pB$.
According to two different sources I've seen the correct answer seems to be (a), which is exactly why I'm trying to understand two things:
Why is (a) a correct answer? Taking two languages in $EXP$, for example, I do not see why it's obvious they are polynomially reducible to each other.
Why is (c) not a correct answer? Taking a language $C\in P$ since $A$ and $B$ are not trivial there are some $a\in A, a'\notin A$ and $b\in B, b'\notin B$, and therefore given a polynomial TM $M$ deciding $C$ I can define a reduction $f$ to $A$ by $f(x)=a$ if $M(x)$ accepts and $f(x)=a'$ otherwise and similarly for $B$, so it seems to me that (c) is a correct answer.