I learned a lot in class today that I’d been mixing up. Who knew I was actually this clueless?
First of all, contrapositive takes an if, then statement (P⇒Q), and makes it into ~Q⇒~P, NOT ~P⇒~Q (this is the inverse).
I also did 9.B.iii. on the homework pretty wrong. The question wants us to prove that if =A ≤ =B and C is any set, then =A ∪ =C ≤ =B ∪ =C. I tried to intuitively prove this by saying that A will map to B, and C will map to itself. But someone else in class brought up the counterexample that if C has a lot more in common with B than with A. That would make the union of B and C smaller than the union of A and B, so there is no one-to-one function that can map from A∪C to B∪C.
Someone else really smart in my class helped me figure out 4.1 #7. I was really confused how to use an injective function and surjective function to prove a third is surjective, but she explained that you can use g∘f being surjective to make g∘f(a) = g(b), because g is injective (which relates the B in g and the B in f), so f(a) = b.
Finally, the POW #8 did not go as I thought it would. S≅ℤ2×ℤ2 does not really mean that these sets are equal, it means that there is a bijective function that makes from S to ℤ2×ℤ2 where they map the same way. This is called an isomorphic relationship. I think I touched on this in my first answer, but didn’t fully explain it. They also have this property where if you add (0,1) and (0,1), you get (0,0), which is similar to taking a contrapositive twice and having it turn the statement back to what it was originally.
And what the heck is with this Kick Hug thing?