https://maximummathematics.com/books/foundational-mathematics/proof/index.htmlIn the previous chapter, we built up the tools of propositional and quantified logic: propositions, connectives, truth tables, the laws of logic, and quantifiers. Along the way, we occasionally ran into the implication — one proposition claiming that another must follow from it — without stopping to give it the attention it deserves. That attention is where we begin this chapter. From there, we turn those tools toward their real purpose: building arguments whose conclusions are guaranteed to be true, and proving that mathematical statements — not just isolated propositions, but general claims about numbers, shapes, and structures — are true beyond any doubt.Hugoen-UShttps://maximummathematics.com/books/foundational-mathematics/proof/a-closer-look-at-the-implication/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/proof/a-closer-look-at-the-implication/index.htmlOut of the two conditional-style connectives we’ve seen — the implication $p \to q$ and the biconditional $p \leftrightarrow q$ — we’ve given the biconditional a fairly thorough treatment already. Now we turn back to the implication, to see what else it has to offer. Recall from its definition that the implication $p \to q$ is false exactly when $p$ is true and $q$ is false. In other words, $\text{true} \to \text{false}$ is a false proposition. This deserves special emphasis:https://maximummathematics.com/books/foundational-mathematics/proof/variations-on-the-implication/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/proof/variations-on-the-implication/index.htmlThere are some simple ways we can change around an implication. Exactly how we make these changes affects how the new implication we form is related to our original starting implication. In this section, we look at three such variations — the converse, the inverse, and the contrapositive — and see how each one relates back to the implication we started with. The Converse, Inverse, and Contrapositive The first two modifications are relatively straightforward.https://maximummathematics.com/books/foundational-mathematics/proof/logical-implications/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/proof/logical-implications/index.htmlSo far, we’ve studied what an implication means on its own, and how it relates to variations like its converse, inverse, and contrapositive. Now we turn to a special kind of implication — one that’s true no matter what truth values its hypothesis and conclusion happen to take on. These implications are especially useful, since knowing one holds lets us deduce its conclusion with total certainty the moment its hypothesis is satisfied.https://maximummathematics.com/books/foundational-mathematics/proof/quantified-logical-implications/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/proof/quantified-logical-implications/index.htmlNow that we’ve taken a closer look at the implication itself, let’s revisit quantified statements to see how the same ideas — logical implication, and the converse, inverse, and contrapositive — carry over to them. A Simple Logical Implication Suppose we have some open statement $p(x)$ with some non-empty universe $\mathcal{U}$ (non-empty just means $\mathcal{U}$ contains at least one element, which we can refer to as $\alpha$).https://maximummathematics.com/books/foundational-mathematics/proof/arguments/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/proof/arguments/index.htmlThe heart of mathematics is not mere computation, but the act of taking a combination of known facts and combining them in some way to arrive at new conclusions. Think back to when you learned the Pythagorean Theorem or the Quadratic Formula. It’s certainly true that these tools help you compute things — the hypotenuse of a right triangle, or the roots of a quadratic function — but that’s a computational activity.https://maximummathematics.com/books/foundational-mathematics/proof/rules-of-inference/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/proof/rules-of-inference/index.htmlWe’ve already seen that, with ever-larger numbers of component propositions, a truth table requires more and more rows to complete — and that logical equivalencies let us simplify compound propositions without needing a truth table at all. We can bypass truth tables when determining whether an argument is valid too. Instead of logical equivalencies, though, we use logical implications. In this section, we collect a list of commonly occurring logical implications, and see how to use them strategically. We’ll still verify each one with a truth table — the point here is just to build up a list of implications we can use later.https://maximummathematics.com/books/foundational-mathematics/proof/using-the-rules-of-inference/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/proof/using-the-rules-of-inference/index.htmlIn the previous section, we collected a large sample of commonly occurring logical implications. We briefly touched on why we’d want such a list — to determine whether a given argument is valid. In addition to determining validity, we can also use the rules of inference to make valid deductions from a given list of premises. In this section, we work through several examples of both use cases. Determining an Argument’s Validity Suppose we’re presented with some argument: a list of premises, and a desired conclusion. We can determine if the argument is valid by appealing to the rules of inference.https://maximummathematics.com/books/foundational-mathematics/proof/logically-equivalent-arguments/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/proof/logically-equivalent-arguments/index.htmlFundamentally, an argument is nothing more than a logical implication — a hypothesis (a conjunction of multiple premises) and a conclusion. We’ve already seen that it’s possible to construct logically equivalent propositions using the laws of logic. Since an argument is fundamentally a proposition based on an implication, it should be possible to construct a different argument that’s logically equivalent to a given one. In some cases, this new, equivalent argument may be easier to verify than the original — which is exactly why it’s worth investing time in constructing logically equivalent arguments in the first place.https://maximummathematics.com/books/foundational-mathematics/proof/invalid-arguments/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/proof/invalid-arguments/index.htmlAll of us have, at one point, been presented with an argument that doesn’t seem quite right. Just because someone can string together a group of premises and assert some conclusion doesn’t mean that conclusion actually follows from the premises. Remember that an argument is valid if the argument’s implication is a logical implication — meaning that no matter what truth values the argument’s propositions have, the overall implication always evaluates to $1$. This means that if we can come up with even one truth value assignment reducing to the form $1 \to 0$, the argument isn’t valid. In other words, for any argument $(p_1 \land p_2 \land \dots \land p_n) \to c$ with $c = 0$ (and all premises true), the argument is invalid.https://maximummathematics.com/books/foundational-mathematics/proof/universal-specification/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/proof/universal-specification/index.htmlThroughout our discussion of arguments so far, we haven’t made use of any quantified statements — though we devoted several sections to quantifiers in the previous chapter, so we certainly got some mileage out of them there. Here, we start to discuss how quantified statements can be used in arguments. The reason we want to do this is that many of the results we’re going to come across are stated in the language of quantifiers. For example, consider the Pythagorean Theorem:https://maximummathematics.com/books/foundational-mathematics/proof/universal-generalization/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/proof/universal-generalization/index.htmlIn the previous section, we talked about a rule of inference that lets us go from broadly true statements to specifically true statements — if something is true for every member of a universe, we can pick out any element from that universe and be assured it still has whatever property we’re interested in. Up to now, none of the arguments we’ve examined have had a universally quantified statement as a conclusion — meaning none of our conclusions could have been generalized. Most results in mathematics are stated in general terms, not specific ones. The Pythagorean Theorem applies to every right triangle, not just isosceles ones, or ones with integer side lengths. The Quadratic Formula doesn’t apply only to $x^2 + 2x + 1 = 0$, or only to $x^2 - 6x + 9 = 0$ — it works even when the leading coefficient isn’t $1$, or when all the coefficients are irrational, or when the corresponding parabola doesn’t even intersect the $x$-axis.https://maximummathematics.com/books/foundational-mathematics/proof/axioms-definitions-theorems-and-proofs/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/proof/axioms-definitions-theorems-and-proofs/index.htmlOver the past two chapters, we’ve been building up a system of mathematical logic — what propositions are, how to determine their truth, when two propositions are equivalent, and how to use propositions in arguments to make valid deductions. However, this isn’t how most of mathematics is communicated. Most of us understand math in terms of numbers and geometric shapes: arithmetic, algebra, trigonometry, lines and angles, polygons and circles, graphs and equations. What we’ve done so far looks very different. What gives?https://maximummathematics.com/books/foundational-mathematics/proof/proof-technique-direct-proofs/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/proof/proof-technique-direct-proofs/index.htmlAt this point, we’ve talked a lot about mathematical logic, arguments, and some common terminology. In this section, we introduce a method for providing a proof for a proposition — the most straightforward technique we have at our disposal. The Underlying Argument Consider a statement such as $p \to q$ — but remember that a theorem is almost always implicitly universally quantified, so what we really want to show is $\forall x\ [p(x) \to q(x)]$ for every element $x$ within some universe $\mathcal{U}$. How would we show this is always true?https://maximummathematics.com/books/foundational-mathematics/proof/proof-technique-indirect-proofs/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/proof/proof-technique-indirect-proofs/index.htmlAs seen in the last section, a direct proof is a proof method where we assume the truth of the hypothesis, and show the truth of the conclusion. But the last example in that section shows that a direct proof can sometimes be quite tricky to devise. If we’re ever stuck trying to show a proposition is a theorem by taking a direct approach, we can use mathematical logic to prove an equivalent implication instead. Since we’re not proving the original implication to be a logical implication, but rather showing a logically equivalent one is, this is called an indirect approach.https://maximummathematics.com/books/foundational-mathematics/proof/proof-technique-contradiction/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/proof/proof-technique-contradiction/index.htmlAs discussed in the previous section, when trying to prove a statement like $p \to q$, we can take an indirect approach by proving some other statement, logically equivalent to $p \to q$, is true. There, the indirect method we used was the contrapositive. In this section, we use the Rule of Contradiction to arrive at another indirect proof method. The Underlying Argument Consider some arbitrary statement $p$. Since the implication $(\neg p \to F_0) \to p$ is always true (as we saw in the section on Rules of Inference), we can write $(\neg p \to F_0) \Longrightarrow p$ — meaning it’s a valid rule of inference, representing the valid argumenthttps://maximummathematics.com/books/foundational-mathematics/proof/mistakes-in-proofs/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/proof/mistakes-in-proofs/index.htmlSo far, we’ve seen three different proof techniques: one direct, and two indirect. Applying any of them requires close adherence to the rules of inference discussed throughout this chapter. However, if we make an argument that uses an invalid inference rule, we have an invalid argument, and hence an invalid proof. In this section, we discuss a few of the most common types of errors that can be made. Violating Hypotheses of a Theorem or Axiom Remember that a theorem guarantees some result holds when a certain collection of premises are satisfied. If even one premise fails to hold in a given scenario, the theorem no longer applies — its conclusion may still happen to be true, but not because of the theorem itself. Consider the following “proof” that $1 = 2$:https://maximummathematics.com/books/foundational-mathematics/proof/proof-technique-equivalence/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/proof/proof-technique-equivalence/index.htmlAll of the proof techniques we’ve discussed so far only seem to go one way. When we provide a proof for $a \Longrightarrow b$, what we’re really saying is that if $a$ is true, then $b$ is true too — but since an implication isn’t generally logically equivalent to its converse, we can’t go the other way: knowing $b$ is true doesn’t necessarily tell us $a$ is also true. However, just because that’s true in general doesn’t mean there are never instances where an implication is logically equivalent to its converse. Consider the statement $n \text{ is even} \Longrightarrow n + 1 \text{ is odd}$. Clearly, its converse is also a logical implication: $n + 1 \text{ is odd} \Longrightarrow n \text{ is even}$. So, whenever “$n$ is even” is true, “$n + 1$ is odd” is also true — and vice versa. These propositions are either simultaneously true, or simultaneously false. Hence, we can write $n \text{ is even} \Longleftrightarrow n + 1 \text{ is odd}$.