I'm reviewing my book Mathematical Proofs a Transition to Advanced Mathematics and looking to understand things at a deeper level. I will try to explain what I've considered so far in regards to this chapter on direct proofs. I'm hoping you guys can point out where I'm correct, incorrect, and any bigger picture ideas I might be missing. Also I'd appreciate any advice you all may have.

Concerning proofs of if/then statements of two open sentences P(x) and Q(x), P(x) --> Q(x), it seems what is really going on is that we're showing the solution set of P(x) is a subset of the solution set of Q(x). It makes sense that this can be done, since a subset with less elements is more restrictive than the larger set it's a part of, and if it meets those restrictions it clearly should also be a member of the larger set. I guess this is an example of the transitive property.

For the if/then statements I've been looking at, many of the premises (P(x)) involve claiming a variable (x, c^2, 5a-7, etc) is an element of some set A, and the conclusions (Q(x)) claim that after running that variable through some function it will become an element of some set B, where A and B don't have to be different. It seems to me that in direct proofs we replace our variable in the conclusion with the requirement to be an element of set A (or I suppose you could say the definition of set A). Then we use algebraic manipulation to turn the function in the conclusion into the requirement to be an element of set B (for example all elements of the even numbers can be expressed in the form 2k where k is an integer). And this would complete the direct proof, showing all solutions for P(x) must also be solutions for Q(x).

As for when to use proof by cases instead of a direct proof, it seems to me to be when set B is a subset of set A. Although the solution sets of P(x) and Q(x) may be equal, it's harder or impossible to directly substitute the requirement for a broader set into a function to prove something about its subset. It's easier to partition that larger set into subsets with higher requirements, and then show that the conclusion is true when those requirements are substituted into the function in place of the variable.

An example for when this would be the case: If t is an integer, then t^2 +3t -1 is odd.

The strategy would be to partition the integers into the set of even and odd numbers, which have higher requirements (E:={2k: k is an integer}. and O:={2k+1: k is an integer}), and prove the function t^2 +3t -1 is odd using a direct proof technique for each subset (case).

I'm curious if every function that transforms a set into its subset has to be proved via proof by cases or if there's an easier (or more complicated i guess) method.