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.
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