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Discrete Structures. Chapter 4: Elementary Number Theory and Methods of Proof 4.1 Direct Proof and Counter Example I: Introduction.

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Discrete Structures

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Discrete structures

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


Assumptions

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.

4.1 Direct Proof and Counter Example I: Introduction


Definitions

Definitions

  • Even Integer

    • An integer n is eveniffn equals twice some integer. Symbolically, if n is an integer, then

      n is even   k  Zs.t. n = 2k.

  • Odd Integer

    • An integer n is oddiffn equals twice some integer plus 1. Symbolically, if n is an integer, then

      n is odd   k  Zs.t. n = 2k + 1.

4.1 Direct Proof and Counter Example I: Introduction


Definitions1

Definitions

  • Prime Integer

    • An integer n is primeiffn > 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 = rsthen either r = 1 and s = n or s = 1 and r = n.

  • Composite Integer

    • An integer n is compositeiffn > 1 and n = rsfor 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 = rsand 1 < r < n and 1 < s < n.

4.1 Direct Proof and Counter Example I: Introduction


Example pg 161 3

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.

      • Is 4rs even?

      • Is 6r + 4s2 + 3 odd?

      • If r and s are both positive, is r2 + 2rs + s2 composite?

4.1 Direct Proof and Counter Example I: Introduction


Disproof by counterexample

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.

4.1 Direct Proof and Counter Example I: Introduction


Example pg 161 13

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.

4.1 Direct Proof and Counter Example I: Introduction


Method of direct proof

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.

4.1 Direct Proof and Counter Example I: Introduction


How to write proofs

How to Write Proofs

  • 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 .

4.1 Direct Proof and Counter Example I: Introduction


Example

Example

  • Prove the theorem:

    The sum of any even integer and any odd integer is odd.

4.1 Direct Proof and Counter Example I: Introduction


Example pg 162 27

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 product of any even integer and any integer is even.

4.1 Direct Proof and Counter Example I: Introduction


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