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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.
Equality between sets can be expressed in terms of subsets. Two sets are equal if and only if they contain each other: that is, A ⊆ B and B ⊆ A is equivalent to A = B. [28] [8] The empty set is a subset of every set: ∅ ⊆ A. [15] If A is a subset of B, but A may not equal to B, then A is called a proper subset of B. This can be written A ...
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 ...
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 ...
For example, {1, 2} is a subset of {1, 2, 3}, and so is {2} but {1, 4} is not. As implied by this definition, a set is a subset of itself. For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B.
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 .
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).
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.