Let \(A\) and \(B\) be two (not necessarily different) sets and let \(f \subseteq A \times B\) be a binary relation. * a total) function (or a map), if it is left-total and right-unique. This is equivalant to saying that for every element \(x\in A\) there is exactly one element \(y\in B\) with \((x,y)\in f\). To express this, we write $f(x)=y$ for functions instead of writing $(x,y)\in f$ as we wrote for general relations. * a partial) function (or a map), if it is right-unique, but not left-total.
The following terms are strongly related to functions: * $A$ is called the domain of $f$. * $B$ is called the codomain of $f$. * The element \(f(x)=y\) for some $x\in A$ and $b\in B$ is called the value of $f$ at the point $x$. * The set \(f[A]:=\{y\in B:f(x)=y\;\text{ for all }x\in A\}\) is called the range: (or image) of $f$. * For some $y\in B$, the set $f^{-1}(y):=\{x\in A:f(x)=y\}$ is called the fiber of $y$ under $f$. * The set $f^{-1}[B]:=\{x\in A:f(x)=y\text{ for all }y\in B\}$ is called the inverse image of $f$. * If a partial function $f$ is undefined for an $a\in A$, i.e. $\not\exists b\in B:f(a)=b,$ then we write $f(a)=\perp.$
Branches: 1
Chapters: 2 3 4
Corollaries: 5 6
Definitions: 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
Examples: 133 134 135 136
Explanations: 137 138 139 140 141
Lemmas: 142 143 144 145 146 147 148
Motivations: 149 150
Parts: 151 152 153
Proofs: 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200
Propositions: 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231
Sections: 232 233
Solutions: 234
Theorems: 235 236 237 238
