Identifying whether two statements are same.

Question

The two statements are:

$\forall a\in A, \exists b\in B, M(a)\implies N(b)$

$(\forall a\in A, M(a))\implies (\exists b\in B, N(b))$

I am not so good in logics. I feel the second statement implies first one. The existence of $b$ is not depending on $a$. So, I think two statements are same.

Am I correct? Can anyone give some counter for sets and predicates if No. Thank you.

Answer 1

Rewriting your two sentences more readably, and adding a crucial pair of parentheses that presumably you had intended: $$\forall a{\in} A\;\exists b{\in}B\;\Big(M(a)\implies N(b)\Big)\tag1$$ $$\forall a{\in} A\;M(a)\implies \exists b{\in} B\; N(b)\tag2$$

1. For each animal, if it is massive, then some boy is neat.
2. If every animal is massive, then some boy is neat.

I feel the second statement implies first one.

In fact, it's the other way round: $(1)$ implies $(2),$ but not vice versa.

Statement $(1)$ guarantees a neat boy as long as there is some massive animal; thus, it is stronger than statement $(2),$ which requires every animal to be massive to guarantee a neat boy.

Answer 2

They are not the same. Consider the following statements:

• $M(a)$: Person $a$ has a dog
• $N(b)$: Person $b$ has a cat

Then the first statement says that for any person, if that person has a cat, then there is a person with a dog. In other words, if there is a cat, then there is a dog.

The second sentence says that if every person has a cat, then someone has a dog.

They are seen to be inequivalent if some but not all people have cats, and no one has a dog.