This document discusses different types of proofs in mathematics including direct proofs, existence proofs, and disproving statements with counterexamples. It provides examples of direct proofs showing that the sum of two even integers is even and that the square of an odd integer is odd. It also gives an example of an existence proof showing there is an even number that can be written as the sum of two prime numbers. Finally, it discusses how to disprove universal statements by finding a counterexample.

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.

4. Proof

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.

9. A Direct Proof of a Theorem

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.

11. A Direct Proof of a Theorem

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.

13. Home Work

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.