CS/Math 240: Introduction to Discrete Mathematics 2/3/2011Lecture 5 : ProofsInstructor: Dieter van Melkebeek Scribe: Dalibor Zelen´yDRAFTUp until now, we have been introducing mathematical notation to capture concepts such aspropositions, implicat i ons, predicates, and sets. We need this machinery in order to be able toargue properties of discrete structures in a rigorous manner. As we were introducing new concepts,we stated various facts and gave proofs of some of them, but we were not explicit about what a goodproof shoul d look like. To day we start discussing what constitutes a valid proof of a propositionand give some guidelines for writing proofs.5.1 ProofsWe briefly mentioned what proofs were in the first lec t ur e. Let’s repeat some of this discussion,and make our definition of a proof more precise.Definition 5.1. A proof of a proposition P is a chain of logical deductions ending in P and startingfrom some set of axioms.Our definition of a proof mentions axioms and logical deductions, both of which require furtherconsideration. Let’s discuss them one by one.5.1.1 AxiomsAxioms are statements we take for granted and do not prove. The set of axioms we use dependson the area we work in. For geometry, we would use Eucl i d’ s five axioms of geometry. Another setof axioms are the ZFC axioms (the abbreviation stands for Zermel o, Fraenkel, and the axiom ofChoice) which form the basis of all set theory. However, both of these sets of axioms are smal l andproving any substantial result starting just from those axioms requires a significant amount of work.Thus, such sets of axioms are more suitable for a course on logic than a course on discrete structures.In this course, we use a much larger set of axioms because our focus is on proof techniques andtheir applications to discr e te structure s. Thus, we will consider any familiar fact fr om math at thelevel of high school as an axiom. If you are unsure whether you can take something for granted onan assignment, just ask.5.1.2 Logical DeductionsLogical deductions, wh i ch are sometimes called inference rules, tell us how to construct proofs ofpropositions out of axioms and other proofs. One example of an inference rule is modus ponens,which says that if we have a proof of P and a proof of P ⇒ Q, then we also have a proof of Q.We now define some terminology and notation for describing inference rules. An inference ruleconsists of antecedents and a conclusion. Antecedents, also known as premises, are axioms or otherproofs. If all t he antece de nts are true, the inference rule says that we have a proof of the conclusion.1Lecture 5: Proofs 5.2. Proof TechniquesWe sometimes also use the word consequence instead of the word conclusion. We illustrate thisterminology on the example of modus ponens.Example 5.1: Modus ponens de al s with two statem ents, P and Q. The antecedents are P andP ⇒ Q, and the conclusion is Q. ⊠To describe inference rules in a more compact way, we draw a horizontal line, place all an-tecedents above the horizontal line (either on the same li ne or on multiple lines), and write theconclusion below the horizontal line. In Figure 5.1a we show the notation for a general inferencerule with antecedent s P1, P2, . . . , Pkand conclusion Q, and we give two ways of writing modusponens in Fi gur e s 5.1b and 5.1c.P1P2. . . PkQ(a) General caseP P ⇒ QQ(b) Modus ponens usingone linePP ⇒ QQ(c) Modus ponens usingtwo linesFigure 5.1: Notation for logical inference rulesAn inferenc e rule is sound if all premises are true and if they i mpl y the conclusion. That is, ifP1, P2, . . . , Pkare premises and Q is the conclusion, we require(P1∧ P2∧ ···∧ Pk) ⇒ Q. (5.1)For modus ponens, the premises are P and P ⇒ Q, so (5.1) in the case of modus ponens says[P ∧ (P ⇒ Q)] ⇒ Q. (5.2)The proposition (5.2) is true only if both P and Q are true. Any other situation m eans the inferenceis not vali d.5.2 Proof TechniquesIn a proof, we apply logical deductions in order to reach the proposition we are proving from a setof axioms. We will not attempt to turn every logical step into the form of Figure 5.1 in this course,and will write proofs at a higher l e vel.We now present some proof techniques. We have seen examples of some of them already, butsome of the proofs were too complicat ed to serve as examples illustrating the proof techniques.Today we present proofs of simpler statements i n order to highlight the pro of techniques us ed .5.2.1 Proving ImplicationsFirst let’s focus on statements of the form P ⇒ Q, i.e., on implications.5.2.1.1 Direct ProofIn order to prove the implication P ⇒ Q using a direct proof, we take the following three steps.Step 1: Assume that P holds. We usuall y write this as the first sentence in the proof.Step 2: Logically derive Q from P .2Lecture 5: Proofs 5.2. Proof TechniquesStep 3: Say that Q holds. This is usually the last sentence in the proof.As an example, we prove the statement that if an integer is odd, then so is its sq uar e . We stateit as Theorem 5.2.Theorem 5.2. ( ∀x ∈ Z) x is odd ⇒ x2is odd.The statements P and Q for the implicati on in Theorem 5.2 are P : “x is odd” and “Q: x2isodd”.Note that the implication is un i versally quantified. Whenever we prove a universally quantifiedstatement, we have to prove it for every x in the domain. In our case, we have to prove theimplication P ⇒ Q for every integer x. It does not suffice to prove, say, that if 3 is o dd, then 9 isodd. We cannot make any assumption about x besides the fact that i t ’ s odd.For the purposes of p re se ntation, we label where the three steps outlined earlier in the margin.Proof of Theorem 5.2.Step 1Let x be an integer, and assume that x is odd.Step 2Since x is odd, we can write x as x = 2y + 1 for some y ∈ Z. In particular, y =x−12. Thenx2= (2y + 1)2= 4y2+ 4y + 1 = 2(2y2+ 2y) + 1. Si nc e y is an integer, so is 2y2+ 2y, which meansthat x2= 2z + 1 for some z ∈ Z.Step 3Therefore, x2is odd.We usually highlight the end of the proof in some way. In the pr oof above, we used a squarein the lower right corner at the end of the last paragraph. Another common way to end a proof isto wri t e Q.E.D. This comes from Latin “quod erat demonstrandum”, which means “which is whathad to be shown”.5.2.1.2 Indirect ProofIn an indirect proof of the implication P ⇒ Q, we prove the contrapositive implication ¬Q ⇒ ¬P .Since the contrapositive of an i m pl i c ati on is logically