Definition 4.1.1 (Integers). An integer is an expression of the form \(a—b\), where \(a\) and \(b\) are natural numbers. Two integers are considered to be equal, \(a—b= c—d\), if and only if \(a + d = c + b\). We let \(\mathbb{Z}\) denote the set of all integers.
Definition 4.1.1
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Exercise 4.1.5
\(=\) { Definition 4.1.1 }
\(\equiv\) { Definition 4.1.1 }
\(\neq\) { By contrapositive of Lemma 2.3.3, \(c \neq 0_N\) and \(d \neq 0_N\), Definition 4.1.1 }
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Exercise 4.1.4
\(=\) { Definition 4.1.1, Commutativity and associativity of the addition on \(\mathbb N\), \((a + c) + b = a + (b + c)\) }
\(=\) { Definition 4.1.1, Commutativity and associativity of the addition on \(\mathbb N\), \(a + (c + b) = (c + a) + b\) }
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Exercise 4.1.3
\(=\) { By Definition 4.1.1, \(a = b — c\) for some natural number \(b\) and \(c\); Definition 4.1.4 }
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Exercise 4.1.2
\(\equiv\) { Definition 4.1.1 }
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Exercise 4.1.1
\(\equiv\) { Definition 4.1.1 }