Let \((a,b),(c,d)\) be ordered pairs of natural numbers. We consider them equivalent, if there exist a natural number \(h\) such that one ordered pair can be obtained from the other ordered pair by adding \(h\) to both natural numbers of that pair, formally
\[(a,b)\sim (c,d)\quad\Longleftrightarrow\quad\begin{cases}(a+h,b+h)=(c,d)& or\(a,b)=(c+h,d+h).\end{cases}]`
The relation "\(\sim\)" defined above an equivalence relation, i.e. for a given ordered pair \((a,b)\in\mathbb N\times\mathbb N\), we can consider a whole set of ordered pairs \((c,d)\in\mathbb N\times\mathbb N\) equivalent to \((a,b)\):
\[x:=\{(c,d)\in\mathbb N\times\mathbb N:\quad( c, d )\sim ( a, b )\}.\]
The set \(x\) is called an integer1. We say that the ordered pair \((a,b)\in\mathbb N\times\mathbb N\) is representing the integer \(x\). The set of all integers is denoted by \(\mathbb Z\).
In order to make a difference in notation, we write \([a,b]\), instead of \((a,b)\), if we mean the integer represented by the ordered pair \((a,b)\) rather than the concrete ordered pair \((a,b)\). A more common (e.g. taught in the elementary school) notation is the notation of integers retrieved from the difference \(a-b\), however, the concept of building a difference is not introduced yet (in fact, we have not introduced the concept of negative integers yet2). For the time being, we give a comparison of the different notations to make more clear:
| Common integer notation | Alternative integer notations | Set of ordered pairs of natural numbers, each notation stands for |
|---|---|---|
| \(\vdots\) | \(\vdots\) | \(\vdots\) |
| \(-3\) | e.g. \([0,3],[1,4],\ldots\) | \(\begin{array}{llllll}\{(0,3),&(1,4),&(2,5),&\ldots,&(h,3+h),&~h\in\mathbb N\}\end{array}\) |
| \(-2\) | e.g. \([0,2],[1,3],\ldots\) | \(\begin{array}{llllll}\{(0,2),&(1,3),&(2,4),&\ldots,&(h,2+h),&~h\in\mathbb N\}\end{array}\) |
| \(-1\) | e.g. \([0,1],[1,2],\ldots\) | \(\begin{array}{llllll}\{(0,1),&(1,2),&(2,3),&\ldots,&(h,1+h),&~h\in\mathbb N\}\end{array}\) |
| \(0\) | e.g. \([0,0],[1,1],\ldots\) | \(\begin{array}{llllll}\{(0,0),&(1,1),&(2,2),&\ldots,&(h,h),&~h\in\mathbb N\}\end{array}\) |
| \(1\) | e.g. \([1,0],[2,1],\ldots\) | \(\begin{array}{llllll}\{(1,0),&(2,1),&(3,2),&\ldots,&(1+h,h),&~h\in\mathbb N\}\end{array}\) |
| \(2\) | e.g. \([1,0],[3,1],\ldots\) | \(\begin{array}{llllll}\{(2,0),&(3,1),&(4,2),&\ldots,&(2+h,h),&~h\in\mathbb N\}\end{array}\) |
| \(3\) | e.g. \([1,0],[4,1],\ldots\) | \(\begin{array}{llllll}\{(3,0),&(4,1),&(5,2),&\ldots,&(3+h,h),&~h\in\mathbb N\}\end{array}\) |
| \(\vdots\) | \(\vdots\) | \(\vdots\) |
Proofs: 1
Algorithms: 1 2 3 4 5
Chapters: 6 7 8 9
Corollaries: 10 11 12 13 14 15
Definitions: 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Examples: 37 38 39 40
Explanations: 41 42 43 44 45
Lemmas: 46 47 48 49
Motivations: 50
Parts: 51 52 53
Problems: 54
Proofs: 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131
Propositions: 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184
Sections: 185 186
Solutions: 187 188
Theorems: 189 190 191
Please note that integers are in fact sets. ↩
The concept of negative integers will be introduced in the discussion of order relation for integers. ↩
