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A is a subset of B (denoted ) and, conversely, B is a superset of A (denoted ). In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B.
A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G). This is often represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e} ). [2] [3] If H is a subgroup of G, then G is sometimes called an overgroup of H.
A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use A ⊂ B and B ⊃ A to mean A is any subset of B (and not necessarily a proper subset), [33] [24] while others reserve A ⊂ B and B ⊃ A for cases where A is a proper subset of B. [31] Examples: The set of all humans is a proper subset of the set ...
If and if is any topological super-space of then is always a (potentially proper) subset of , which denotes the closure of in ; indeed, even if is a closed subset of (which happens if and only if = ), it is nevertheless still possible for to be a proper subset of .
If A is a subset of B, then one can also say that B is a superset of A, that A is contained in B, or that B contains A. In symbols, A ⊆ B means that A is a subset of B, and B ⊇ A means that B is a superset of A. Some authors use the symbols ⊂ and ⊃ for subsets, and others use these symbols only for proper subsets. For clarity, one can ...
2. A proper subset of a set X is a subset not equal to X. 3. A proper forcing is a forcing notion that does not collapse any stationary set 4. The proper forcing axiom asserts that if P is proper and D α is a dense subset of P for each α<ω 1, then there is a filter G P such that D α ∩ G is nonempty for all α<ω 1
For instance, had been declared as a subset of , with the sets and not necessarily related to each other in any way, then would likely mean instead of . If it is needed then unless indicated otherwise, it should be assumed that X {\displaystyle X} denotes the universe set , which means that all sets that are used in the formula are subsets of X ...
Any family of subsets of a set is itself a subset of the power set ℘ if it has no repeated members. Any family of sets without repetitions is a subclass of the proper class of all sets (the universe ).