Section 1.1 Introduction to mathematical arguments
To prepare for class, we read (most of) the first portion of “Introduction to mathematical arguments” by Michael Hutchings, which is posted on Canvas. Here are some of the things in the reading I want to emphasize or say more about.
- If…then
- We need some more vocab here. When considering ‘if \(p\text{,}\) then \(q\)’, we call \(p\) the hypothesis and \(q\) the conclusion. Critical to our understanding later is that there is exactly one circumstance in which an if…then statement (sometimes called an implication or a conditional statement) is false: the hypothesis \(p\) is true and the conclusion \(q\) is false.
- If and only if
- This sort of statement is also called a biconditional statement. We usually prove \(p\Leftrightarrow q\) by proving \(p\Rightarrow q\) and separately that \(q\Rightarrow p\text{.}\)
- Negating statements
- We could get really deep into the weeds of something called truth tables for this, but to be honest, we just don’t have the time for that. (It’s also excruciatingly dull to do it.) However, I don’t want some of these things to seem magical. One way to approach the negation of a statement is that the negation is true in exactly the circumstance(s) where the original statement is false. Let’s take a look at a couple of important ones.
- \(p\) and \(q\text{:}\) This is true exactly when both are true. Thus, this is false when at least one statement is false. Hence, the negation is true when at least one statement is true. To shuffle the “not” around a bit, that means we need at least one of “not \(p\)” and “not \(q\)” to be true, so the negation is (not \(p\)) or (not \(q\)).
- \(p\) or \(q\text{:}\) This is true when at least one is true. Thus, this is false exactly when both statements are false. In this case, both “not \(p\)” and “not \(q\)” are true, so the negation is (not \(p\)) and (not \(q\)).
- \(p\Rightarrow q\text{:}\) This false exactly when \(p\) is true and \(q\) is false. Thus, the negation is true in exactly this situation, which requires that both \(p\) and (not \(q\)) be true, so \(p\) and (not \(q\)) is the negation. We will see why this is important when we do proof by contradiction later on.
- A note about quantifier symbols
- In this class, you will never use the symbols \(\exists\) or \(\forall\) in work submitted for grading. Never. If you do, you will not get full credit for the problem. As far as I am concerned, there are two settings where these symbols are useful: in courses in mathematical logic (MATH 571, for instance) and in scratch work where a complicated statement is being negated. Once the scratchwork is done, you carefully write things using the proper words.
- Theorem statements
- A statement that is true and for which we have a proof is a theorem. As we go on, we will talk about related words such as “proposition”, “lemma”, and “corollary”, but those are all words that describe the role of a theorem. You can’t go wrong by calling a true statement a thorem, so we will stick with that for the time being. Hutchings’s essay somewhat silently transitions from statements of theorems that look like “For every integer \(x\text{,}\) if \(x\) is odd, then \(x+1\) is even.” to theorem statements that look like “If \(x\) is an odd integer, then \(x+1\) is even.” When you see a theorem that appears to have a variable that doesn’t have a quantifier attached, such as \(x\) in my second example, you assume a “for all” quantifier on that variable out in front of the implication. Mainly a convenience of mathematical writing.
