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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. The relationship of one set being a subset of another is called inclusion (or sometimes containment).
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 ...
A subset of having elements is called a -subset of . The -subsets () of a set form a family of sets. Let = {,,,,}. An example of a family of sets over (in the ...
The first condition states that the whole set B, which contains all the elements of every subset, must belong to the nested set collection. Some authors [ 1 ] do not assume that B is nonempty. The second condition states that the intersection of every couple of sets in the nested set collection is not the empty set only if one set is a subset ...
A subset of is a regular open set if and only if its complement in is a regular closed set. [2] Every regular open set is an open set and every regular closed set is a closed set . Each clopen subset of X {\displaystyle X} (which includes ∅ {\displaystyle \varnothing } and X {\displaystyle X} itself) is simultaneously a regular open subset ...
A bijection between two topological spaces is a homeomorphism if and only if the derived set of the image (in the second space) of any subset of the first space is the image of the derived set of that subset. [7] A space is a T 1 space if every subset consisting of a single point is closed. [8]
In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are.
The 5th roots of unity in the complex plane form a group under multiplication. Each non-identity element generates the group. In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses.