4.1 Direct Proof and Counter Example I: Introduction

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Discrete Structures
Chapter 4: Elementary Number Theory and Methods of
Proof
4.1 Direct Proof and Counter Example I: Introduction
Mathematics, as a science, commenced when first someone, probably a
Greek, proved propositions about “any” things or about “some” things
without specification of definite particular things.
– Alfred North Whitehead, 1861-1947
4.1 Direct Proof and Counter Example I:
Introduction
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Assumptions
We assume that
• we know the laws of basic algebra (see Appendix A).
• we know the three properties of equality for objects A,
B, and C:
 A=A
 If A = B then B = A
 If A = B,B = C, then A = C
• there is no integer between 0 and 1 and that the set of
integers is closed under addition, subtraction, and
multiplication.
• most quotients of integers are not integers.
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Definitions
• Even Integer
– An integer n is even iff n equals twice some
integer. Symbolically, if n is an integer, then
n is even   k  Z s.t. n = 2k.
• Odd Integer
– An integer n is odd iff n equals twice some integer
plus 1. Symbolically, if n is an integer, then
n is odd   k  Z s.t. n = 2k + 1.
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Definitions
• Prime Integer
– An integer n is prime iff n > 1 for all positive integers r
and s, if n = rs, then either r or s equals n Symbolically, if
n is an integer, then
n is prime   r, s  Z+ , if n = rs then either r = 1 and s =
n or s = 1 and r = n .
• Composite Integer
– An integer n is composite iff n > 1 and n = rs for all
positive integers r and s with 1 < r < n and 1 < s < n.
Symbolically, if n is an integer, then
n is composite   r, s  Z+ s.t. n = rs and 1 < r < n and
1 < s < n.
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Example – pg. 161 # 3
• Use the definitions of even, odd, prime, and
composite to justify each of your answers.
– Assume that r and s are particular integers.
a. Is 4rs even?
b. Is 6r + 4s2 + 3 odd?
c. If r and s are both positive, is r2 + 2rs + s2 composite?
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Disproof by Counterexample
• 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.
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Example – pg. 161 # 13
• Disprove the statements by giving a
counterexample.
– For all integers m and n, if 2m + n is odd then m
and n are both odd.
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Introduction
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Method of Direct Proof
• Express the statement to be proved in the form
“ x  D, if P(x) then Q(x).”
• Start the proof by supposing x is a particular
but arbitrarily chosen element of D for which
the hypothesis P(x) is true. (Abbreviated:
suppose x  D and P(x).)
• Show that the conclusion Q(x) is true by using
definitions, previously established results, and
the rules for logical inference.
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Introduction
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How to Write Proofs
1.
2.
3.
4.
5.
6.
7.
8.
9.
Copy the statement.
Start your proof with: Proof:
Define your variables.
Write your proof in complete, grammatically
correct sentences.
Keep your reader informed.
Given a reason for each assertion.
Include words or phrases to make the logic clear.
Display equations and inequalities.
Conclude with .
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Example
• Prove the theorem:
The sum of any even integer and any odd
integer is odd.
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Introduction
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Example – pg 162 # 27
• Determine whether the statement is true or
false. Justify your answer with a proof or a
counterexample as appropriate. Use only the
definitions of terms and the assumptions on
page 146.
– The sum of any two odd integers is even.
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Introduction
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