For example, a well-known function due to Leonhard Euler is deﬁned by theformulaf(n) =n2+n+41 for all integersn. If you substitute the numbersn=0,1,2,...,39 into this function, you obtain the numbers 41, 43, 47,..., 1601, allof which are prime numbers. It therefore might appear that substituting in everypositive integer into this function would result in a prime number (which would bea very nice property), but it turns out thatf(40) =1681=412, which is not prime.See [Rib96, p. 199] for more discussion of this, and related, functions. The point isthat if you want to prove that a statement is true for allxinU, it does not sufﬁce totry only some of the possible values ofx.Statements of the form(∀xinU)P(x)can be proved by strategies other than di-rect proof. For example, the proof of such a statement using proof by contradictiontypically has the following form.Proof.We use proof by contradiction. Lety0be inU. Suppose thatP(y0)is false....(argumentation)...Then we arrive at a contradiction.ut
722 Strategies for ProofsWe will not show here any examples of proofs of statements of the form(∀xinU)P(x), because we have already seen a number of such proofs in the pre-vious sections of this chapter.We now consider statements with a single existential quantiﬁer, that is, statementsof the form “(∃xinU)P(x).” Using the Existential Generalization rule of inference inSection 1.5, we see that to prove a theorem of the form(∃x)P(x)means that we needto ﬁnd somez0inUsuch thatP(z0)holds. It does not matter if there are actuallymanyxinUsuch thatP(x)holds; we need to produce only one of them to proveexistence. A proof of “(∃xinU)P(x)” can also be viewed as involving a statementof the formA→B. After we produce the desired objectz0inU, we then prove thestatement “ifx=z0, thenP(x)is true.” Such a proof typically has the following form.Proof.Letz0=.......(argumentation)...Thenz0is inU....(argumentation)...ThenP(z0)is true.utHow we ﬁnd the elementz0in the above type of proof is often of great interest,and sometimes is the bulk of the effort we spend in ﬁguring out the proof, but it isnot part of the actual proof itself. We do not need to explain how we foundz0inthe ﬁnal write-up of the proof. The proof consists only of deﬁningz0, and showingthatz0is inU, and thatP(z0)is true. It is often the case that we ﬁndz0by goingbackwards, that is, assuming thatP(z0)is true, and seeing whatz0has to be. However,this backwards work is not the same as the actual proof, because, as we shall see,not all mathematical arguments can be reversed—what works backwards does notnecessarily work forwards.We now turn to a simple example of a proof involving an existential quantiﬁer.Recall the deﬁnitions concerning 2×2 matrices prior to Theorem 2.4.5. We say thata 2×2 matrixM=(a bc d)has integer entries ifa,b,canddare integers.