Discrete Maths: First Order Logic - Question in my mind based on question from Kenneth Rosen

Question

This is not a direct question from Rosen but a question that popped up in my head as I was solving problems from Rosen.

1) ∀_x∃_y(x≠y → M(x,y))

2) ∀_x∃_y(x≠y ∧ M(x,y))

Here the domain of x,y is the students in your class.  M(x,y) represents the statement x has sent y an email.

My question is: Are these two statements equivalent or do they mean different things?

Also, could you please explain the difference between the meaning of these statements (in plain English) if they are different?

Answer 1

∀_x∃_y(x≠y → M(x,y)) says that "if sender and receiver are different then send mail" that means sending of mail depends sender and receiver

1. if sender and receiver are different and mail also send then above statement is correct.

2. if sender and receiver are different and mail not sent then above statement is not correct.

3. if first condition is itself false then no meaning of looking for second condition that means statement is true.

∀_x∃_y(x≠y ∧ M(x,y)) says that "sender and receiver are different and  send mail" that means sending of mail does not depends on sender and receiver, both conditions shold be true for making statement true.

so ∀_x∃_y(x≠y → M(x,y)) and ∀_x∃_y(x≠y ∧ M(x,y)) are different things.

Comments

Great Explanation Sir!!!