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Discrete Structures
Chapter 4: Elementary Number Theory and Methods of
Proof
4.2 Direct Proof and Counter Example II: Rational
Numbers
Such, then, is the whole art of convincing. It is contained in two
principles; to define all notations used, and to prove everything by
replacing mentally defined terms by their definitions.
– Blaise Pascal, 1623-1662
4.2 Direct Proof and Counter Example II:
Rational Numbers
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Definitions
It is important to be aware that there are a number of words that mean
essentially the same thing as the word “theorem,” but which are used in
slightly different ways.
• Theorem
– In general the word “theorem” is reserved for a statement that is
considered important or significant (the Pythagorean Theorem, for
example).
• Proposition
– A statement that is true but not as significant is sometimes called a
proposition.
4.2 Direct Proof and Counter Example II:
Rational Numbers
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Definitions
It is important to be aware that there are a number of words that mean
essentially the same thing as the word “theorem,” but which are used in
slightly different ways.
• Lemma
– A lemma is a theorem whose main purpose is to help prove another
theorem.
• Corollary
– A corollary is a result that is an immediate consequence of a theorem
or proposition.
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Rational Numbers
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Definitions
• Rational Number
– A real number r is rational iff it can be expressed
as a quotient of two integers with a nonzero
denominator. More formally, if r is a real number,
then
r is rational   a, b  Z s.t. r = a/b and b  0.
• Irrational Number
– A real number that is not rational is irrational.
4.2 Direct Proof and Counter Example II:
Rational Numbers
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Zero Product Property
• If neither of two real numbers is zero, then
their product is also not zero.
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Theorems
• Theorem 4.2.1
– Every integer is a rational number.
• Theorem 4.2.2
– The sum of any two rational numbers is rational.
• Corollary 4.2.3
– The double of a rational number is rational.
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Example – pg. 169 # 15, 16, 18
• Determine whether the statements are true or false. Prove
each true statement directly from the definitions, and give
a counterexample for each false statement. In case the
statement is false, determine whether a small change
would make it true. If so, make the change and prove the
new statement.
15. The product of any two rational numbers is a rational number.
16. The quotient of any two rational numbers is a rational number.
18. If r and s are any two rational numbers, then r  s is rational.
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Example – pg. 169 # 21
• Use the properties of even and odd integers that are
listed in example 4.2.3 to do exercise 21. Indicate
which properties you use to justify your reasoning.
– True or False? If m is any even integer and n is and odd
integer, then m2 + 3n is odd. Explain.
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Example – pg. 169 # 24
• Derive the statement below as corollaries of theorems
4.2.1, 4.2.2, and the results of exercise 15.
– For any rational numbers r and s, 2r + s is rational.
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Example – pg. 169 # 28
• Suppose a, b, c, and d are integers and a  c.
Suppose also that x is a real number that satisfies the
equation
ax  b
1
cx  d
Must x be rational? If so, express x as the ratio of two
integers.
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Example – pg. 169 # 30
• Prove that if one solution for a quadratic equation of
the form x2 + bx + c = 0 is rational (where b and c are
rational), then the other solution is also rational.
Hint: Use the fact that if the solutions of the equation
are r and s, then x2  bx  c   x  r  x  s 
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Rational Numbers
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