Hmmmm

There was only 1 lecture so far this week because of thanks giving, and there were no tutorials. So I'm not sure what to write about. But I was thinking about something someone said in tutorial, and I thought of an answer this week, and I think it's interesting, so I will write about it.

Someone said that when you do ∃x, y, it should automatically mean x is different from y, because whenever you write ∃x, y you don't want them to be the same thing. It seemed right, but sometimes you don't care if x and y are the same. I thought of this example this week.

I say "The set S of positive even numbers excluding four is closed under addition." You say that's not true, and there's a counterexample: ∃x, y ∈ S, ¬(x + y ∈ S). But I don't think there are counterexamples if x and y are different. But if x and y are the same, 2 + 2 is 4 which is not in S.

So, sometimes when we use ∃x, y, we don't care if x and y are the same. Actually, in this example, it's important that x and y are generic.

This looks kinda short, so I will write about how one of my other classes related to CSC165. There was a question in a past midterm that said "the logic function f is true when any 2 of the inputs are true." My friend and I thought it meant "at least two are true", because if three inputs are true, there are "any 2" inputs that are true. But in the answer, "any" meant "only". In English, people might argue about exactly what things mean. But if I write it symbolically, I think most people would agree that "any two are true" is ∃x, y ∈ S, x not equal to y, x ∧ y. Now, it's clear that if three inputs are true, "any two are true" is still true. It's also clear that P(S) = ∃x, y, z ∈ S, x not equal to y not equal to z, x ∧ y ∧ z is a subset of Q(S) = ∃x, y ∈ S, x not equal to y, x ∧ y (the set of sets such that P holds is a subset of the set of sets such that Q holds). If P is a subset of Q, then P ⇒ Q. If Q(S), then we know three different inputs x, y, z are true. I can pick two for satisfying Q(S). Symbolic notation is good