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| Wut. |
I keep looking at the assignment and my mind is going blank, this week is going to be soooo fun. On another note, I've observed that I'm making posts on later and later days of the week as I continue to delay at a blistering pace. Procrastination is not fun guys, let this be a lesson to all of you. I'd say that I'll try to make them sooner, but lets be honest would I?
Now that I'm working on the assignment, there are definitely tons of questions on my mind (how convenient that they weren't there earlier). One question I have, is if there is a statement that I know a counter example for, can I state the counterexample and reach the conclusion that P(x) -> notQ(x) for all values? Will that be enough to invalidate the entire statement? Also if I have a statement that has a for all assertion in the consequent, what on earth do I do? For example 
x in D, P(x) -> ( y in D, Q(y) ^ P(y)), is there anything I can do to solve this or am I completely boned and should view the statement from a different angle?
x in D, P(x) -> ( y in D, Q(y) ^ P(y)), is there anything I can do to solve this or am I completely boned and should view the statement from a different angle?
What is really irking me right now is the fact that the course notes are mocking me. The two step method it gives to solving proofs seems pretty understandable, but it states in bold IF YOU DON'T UNDERSTAND WHY SOMETHING IS TRUE, DON'T EXPECT TO PROVE IT! In my opinion that little bit of advice is just like saying "if you don't know how to prove it, don't expect to prove it". You don't say? Well then, that'll definitely square me off on my problem solving journey. I cannot expect to fail now that I know that the secret to solving proofs is to understand them.
Scepticism aside, now that I've started looking at other blogs (the few that make an effort of weekly posts, you go giiirl!), it seems that some people are really getting into the groove with these proofs. A lot of people seem to be done with the assignment already, and how they accomplished this Herculean fear was just by sitting down for a few hours and working hard and getting it done. Now it's got me thinking I should have a more optimistic view of the proofs. I just gotta sit down for several hours, enjoy what I'm doing and just power through it. I think I'm going to try really really really hard and make myself drink the CSC 165 kool aid this week. Embrace the proof, and become one with it.
Good luck on the assignment everyone, may Heap be with you.
I have the same question about if there's a statement that you have a counter example for, if you can just state the counter example and show that the statement is false. I haven't been able to find anywhere in the Course Notes where it says that you can do that, but surely since we spent so much time learning that "to disprove a universal quantifier, you need to show one counter example", we should be able to do that.
ReplyDeleteLet me know if you find out.
Say you have to prove "For all x in N, y in N, x == y", which is obviously false. The disproof would go like this:
ReplyDeleteDisprove "For all x in N, y in N, x == y", so prove the negation, which is "There exists an x in N, y in N, x != y".
Pick x = 5
Then x in N
Pick y = 6
Then y in N
Then x != y
Then there exists an x in N, y in N, x != y
Which is sufficient to disprove the statement using a counterexample.