Understanding Proof Writing: Why We Use 3) instead of 4)

[Salt]

I have no idea how to type math symbols into here so it's all in the PNG attached.

I'm probably kind of dumb for not getting this but...

I understand that 1) & 3) are true. And the 2) is not right, as it means all x are members of F and true for P(x) when we mean all x that are members of F are true for P(x).
But why do we use 3) instead of 4)?

 

[HallsofIvy]

(4) is not always a true statement. The right hand side of (4) would be true even if F were empty whereas the left hand side would not be. Notice that if x is NOT in F then "x contained in F implies P(x)" is a TRUE statement because the hypothesis is FALSE.

matt, that was pretty much what you said. Why did you delete it?

 

[matt grime]

Cos when I looked more closely I decided that I couldn't decipher the small subscript on the LHS with any certainity.

 

[Salt]

Thanks everyone. Sorry about the size, I attached a bigger one in this post.

So from what I understand from reading the replies and scratching my head over the AND and IMPLIE truth tables.

right side of 3) asserts :

• there exist a x such that it's a member of F and true for P(x)

right side of 4) asserts :

1. there exist a x such that it's a member of F and true for P(x) , or
2. there exist a x such that it's NOT a member of F and true for P(x) , or
3. there exist a x such that it's NOT a member of F and NOT true for P(x)

However we do not wish to state as true 2. and 3. , for it would implie that there exist a x that is NOT a member of F. As the set representing "not F" may or may not be empty.

Anyway that's the reasoning I manage to arrive at.

 

[matt grime]

A=>B is precisely "B or not(A)".