Proof– An axiom is a proposition that is simply accepted as true.– A proof is a sequence of logical deductions from axioms and

previously-proved statements that concludes with the proposition in question.

– Logical deductions or inference rules are used to prove new propositions using previously proved ones.

Proof

Basic Definitions

oAn integer n is even if and only if n is twice some integer k.– n is even an integer k such that n = 2k.⇔ ∃

o An integer n is odd if and only if n is twice some integer k plus 1.– n is odd an integer k such that n = 2k + 1.⇔ ∃

– Is -461 odd?

– If a and b are integers then is 4a + 10b is even? Yes 2(2a + 5b)

Yes 2(−151) + 1.

Direct Proofo The implication p q can be proved by showing that if p is

true, the q must also be true.

o This shows that the combination p true and q false never occurs. A proof of this kind is called a direct proof.

Method of Direct Proof1. Express the statement to be proved in the form

“ x D, if P(x) then Q(x).” ∀ ∈ (This step is often done mentally.)

2. Start the proof by supposing x is a particular but arbitrarily chosen element of D for which the hypothesis P(x) is true. (This step is often abbreviated “Suppose x D and P(x).”)∈

3. Show that the conclusion Q(x) is true by using definitions, previously established results, and the rules for logical inference.

A Direct Proof of a Theoremo Prove that the sum of any two even integers is even.o Formal Restatement: integers ∀ m and n, if m and n are even

then m + n is even.o Starting Point: Suppose m and n are particular but arbitrarily

chosen integers that are even.o To Show: m + n is even.o If the existence of a certain kind of object is assumed or has been deduced then it can be given a

name, as long as that name is not currently being used to denote something else.

A Direct Proof of a Theorem

Direct Proof and Counterexampleo Give a direct proof of the theorem “If n is an odd integer, then n2 is odd.”

∀n P(n) → Q(n), where P(n) is “n is an odd integer” and Q(n) is “n2 is odd.”

By def. n = 2k + 1, where k is some integer.

To prove n2 is odd, take square of both sidesn2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1

(it is one more than twice an integer).

Consequently, we have proved that if n is an odd integer, then n2 is an odd integer.

A Direct Proof of a Theorem

Direct Proof and Counterexampleo Prime & Composite

– 6=2· 3 is a product of two smaller positive integers– 7=1.7

– A positive integer that cannot be written as a product of two smaller positive integers is called prime.

Home Work

Proving Existential Statements∃x D such that Q(x)∈

is true if, and only if,

Q(x) is true for at least one x in D.

1. find an x in D that makes Q(x) true.

2. Give a set of directions for finding such an x.

o Both of these methods are called constructive proofs of existence.

Proving Existential Statementso Prove the following: an even integer n that can be written ∃

in two ways as a sum of two prime numbers.o Let n = 10. Then 10 = 5 + 5 = 3 + 7 and 3, 5, and 7 are all prime

numbers.

o Suppose that r and s are integers. Prove the following: an ∃integer k such that 22r + 18s = 2k.o Let k = 11r + 9s. o Then k is an integer because it is a sum of products of integers;

o and by substitution, 2k = 2(11r + 9s), which equals 22r + 18s by the distributive law of algebra.

Proving Existential Statementso A nonconstructive proof of existence involves showing

eithera) that the existence of a value of x that makes Q(x) true is

guaranteed by an axiom or a previously proved theorem or

b) that the assumption that there is no such x leads to a contradiction.

o The disadvantage of a nonconstructive proof is that it may give virtually no clue about where or how x may be found.

Disproving Universal Statements by Counterexampleo To disprove a statement means to show that it is false.

∀x in D, if P(x) then Q(x).

– Showing that this statement is false is equivalent to showing that its negation is true

∃x in D such that P(x) and not Q(x).

– Example is given to show that statement is true and actual statement is false.

– Such as example is called counterexample.

Disproving Universal Statements by Counterexampleo To disprove a statement of the form “ x D, if P(x) then Q(x),” ∀ ∈

find a value of x in D for which the hypothesis P(x) is true and the conclusion Q(x) is false. Such an x is called a counterexample.

o Statement: real numbers a and b, if a∀ 2 = b2, then a = b.o Counterexample:o Let a = 1 and b = −1. Then a2 = 12 = 1 and b2 = (−1)2 = 1, and

so a2 = b2. But a b since 1 −1.