3. 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.
5. 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.
6. Direct Proof
o 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.
7. Method of Direct Proof
1. 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.
8. A Direct Proof of a Theorem
o 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.
10. Direct Proof and Counterexample
o 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 sides
n2 = (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.
12. Direct Proof and Counterexample
o 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.
14. 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.
15. Proving Existential Statements
o 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.
16. Proving Existential Statements
o A nonconstructive proof of existence involves showing
either
a) 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.
17. Disproving Universal Statements by
Counterexample
o 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.
18. Disproving Universal Statements by
Counterexample
o 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 a2 = 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.
