Is $\exists a \in A. \forall b \in B \Rightarrow P(a, b)$ equivalent to $\forall c \in B. \exists d \in A \Rightarrow P(c, d)$?

Question

Given 2 statements:

$$\exists a \in A. \forall b \in B \Rightarrow P(a, b)$$ and $$\forall c \in B. \exists d \in A \Rightarrow P(d, c)$$

where lowercase letters are elements(or items); A and B are sets that define the elements; P is a relationship between the two items in the bracket.

Are they equivalent to each other? And Why?

Answer 1

No.

It is not true that there exists a woman on Earth (a in A) such that for every man on Earth (b in B), the woman is his mother [P(a,b)].

However, it is true that for every man on Earth (c in B) there exists a woman on Earth (d in A) such that the woman is his mother [P(c,d)].