I'm not sure what I'm being asked to do in Problem 1.8.3. Should I prove "(P∧¬Q)→0" by cases?
No, you are asked to give a direct proof of "P→Q", which means that you should assume P and prove Q. To make this proof a proof by cases, you need to pick some proposition R and then both (1) assume "P∧R" and prove Q, and (2) assume "P∧not;R" and prove Q. You are given that you know how to prove 0 from "P∧¬Q" -- using this proof and other valid rules, tell me how you can prove both (1) and (2) with the R of your choice.
I am confused by the solution to Exercise 1.8.1. In the first line, how can we assume "0∧p" when it is obviously false? And don't the second and fourth lines contradict each other?
It is true that no assignment of truth value to p can make "0∧p" true. But we can still assume it for the purposes of making a direct proof of the implication "(0∧p)→0", which is what we want to prove here. The second line of the proof is part of a "fantasy world" where "0∧p" is assumed to be true. By showing that "0" is true in this fantasy world, we complete the direct proof. Line 4, on the other hand, is in the real world where "0" is false. So there is no conflict with line 2, which has an assumption in force.
In Problem 1.5.2, the "and" symbol looks a little strange. Can I take the "and" of two sets?
No, you can't. The "and" (∧) is a mistake and should be an "intersection" (∩) symbol.
Can I show a counterexample with Venn diagrams in Problem 1.5.2?
As with diagrams in geometry, Venn diagrams can illustrate a proof or an example but don't really give the example or proof itself. A good way to use Venn diagrams in this problem is as follows. Suppose the statement were "A ∩ B = A ∩ C". You draw a Venn diagram and find that the region "A ∩ B ∩ C-bar" is in one set but not the other. So any three sets, where some element x is in A and B but not in C, will work. The simplest example is A = B = {x} and C = ∅ so A ∩ B = {x} and A ∩ C = ∅ and the statement isn't true.
Last modified 8 February 2005