Let \(X\) be a set. A set \(A\) is called a subset of \(X\) and denoted by \(A\subseteq X\), if each element of \(A\) is also an element of \(X\). Equivalently, X is a superset of \(A\), denoted as \(X\supseteq A\).
If, in addition, \(A\neq X\), then \(A\) is called a proper subset of \(X\) and denoted by \(A\subset X\). Equivalently, X is then a proper superset of \(A\), and denoted as \(X\supset A\).
We can draw supersets and subsets as Venn diagrams. In the following figure, $B$ is a subset of $A$ and $A$ is a subset of the universal set $U$:
Axioms: 1
Corollaries: 2 3 4 5 6 7 8 9 10 11
Definitions: 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
Examples: 116 117 118
Explanations: 119 120 121 122 123 124
Lemmas: 125 126 127 128 129
Motivations: 130 131
Parts: 132 133
Problems: 134
Proofs: 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205
Propositions: 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241
Sections: 242
Solutions: 243
Theorems: 244 245 246 247 248 249 250
