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4
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In this case, we write a | b
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A Restatement
Theorem 3
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Multiple and Divisor Description
If a | b, then we also say that b is a multiple of a and that a is a divisor
of b
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Examples
Result 4
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If a | b and b | c,
then a | c
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Then b = ax and c = by, where x, y ∈ ℤ
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Since xy is an integer, a | c
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4 Let x ∈ ℤ
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Assume that 2 | (x² - 1)
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Thus, x² =
2y + 1 is an odd integer
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12 that x too is
odd
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Then,

x² - 1 = (2z + 1)² - 1 = (4z² + 4z + 1) - 1 = 4z² + 4z = 4(z² + z)
Since (z² + z) ∈ ℤ, 4 | (x² - 1)
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6 Let x ∈ ℤ
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(b) If x is even, then 4 | x²; while if x is odd, then 8 | (x² - 1) and
so 4 | (x² - 1)
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6(a):
Assume, first, that 3 | x
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Hence, x² =
(3q)² = 3(3q²)
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Next, suppose that
3 ∤ x
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We
consider these two cases
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6(b):
First, assume that x is even
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Thus, x² =
(2q)² = 4q²
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Next, suppose that x is odd
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We consider
these two cases
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2 Proofs Involving Congruence of Integers
Background Knowledge
An integer is even if it's in the form 2q and odd if it's 2q + 1
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Novel Knowledge
Integers can be expressed as 3q, 3q + 1, or 3q + 2 based on their
remainder when divided by 3
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The concept of congruence is when one integer is similar to another
modulo a third
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Modulo Notation
Every integer can be represented in forms related to division by 2 or 3
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When divided by 4, every integer x will be congruent to 0, 1, 2, or 3
modulo 4
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This pattern of congruence can be extended for divisions by any
integer n that's 5 or greater
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3 Proofs Involving Real Numbers
Properties of Real Numbers
1
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2
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3
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4
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- When c > 0, the relation a/c ≥ b/c is true if a ≥ b, and both ac >
bc and a/c > b/c hold when a > b
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5
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4
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The intersection of sets A and B is denoted by A ∩ B, which includes
all elements that are common to both A and B
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The union of sets A and B is represented by A ∪ B, consisting of
elements that are in A, in B, or in both
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The difference between sets A and B is represented by A - B, which
consists of elements that are in A but not in B
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The relative complement of B in A is A - B
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5
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6
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1, provide visual representations of set
operations such as A - B and A ∩ B
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7
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To prove the inclusion C ⊆ D, one has to
show that every element in C is also in D
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If set C has no elements, it is the empty set {∅}, and is always a
subset of D
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5 Fundamental Properties of Set Operations

Preface
Key results about sets are derived from fundamental set properties,
which stem from logical statements discussed in Chapter 2
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4
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If
either A or B is an empty set, then the Cartesian product is also empty
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