(CSC165)_Oct 24_More About Proof

After we learned the general structure of proof, professor began to introduce some example of actually proving some statement to help us build the logical connection. Actually, when writing proof, it is a lot similar to what we learned about 1) problem solving and 2) logic.

Sometime we need to decide if we want to prove the statement or disprove it. To prove it, we can use direct prove or indirect prove. Direct prove are more like doing logical analysis step by step after assuming the antecedent and finally to arrive at the conclusion. Indirect prove is very useful when its hard to find the connection from a direct way, and, in this case, we can prove the original statement by proving its contrapositive. Since, we know a statement's truth value is the same as its contrapositive. I think after more practice, the idea thinking pattern of trying to prove a statement is to think through both side, and being able to figure out which is a more efficient way to write a proof.

In terms of disprove, some of us have questions about how we could know the statement was wrong in the first place. Actually, most of the questions we have at this moment is very easy to find out if it is wrong by simply plugging some examples. However, when facing with a more complex statement, the professor told us, which I think is quite interesting but probably very important, that when we tried to prove a statement, and stuck somewhere during the process, we may think backwards and also think about disproving it. Basically, we can prove the original statement's converse to disprove the statement. Alternatively, if we want to do the disprove directly, we can also assume the antecedent, and after that if we can logically find a contradiction, we also achieve our goal.