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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 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 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 ...
For example, if g: Y → Z is a proper morphism of locally noetherian schemes, Z 0 is a closed subset of Z, and Y 0 is a closed subset of Y such that g(Y 0) ⊂ Z 0, then the morphism ^: / / on formal completions is a proper morphism of formal schemes.
It would be useful if the article explains or defines what the proper subset and superset *is* before introducing the symbols for them. 86.12.162.37 ( talk ) 16:32, 12 January 2018 (UTC) [ reply ] This is done in the section on definitions, which comes before the section on this notation.
A simple example is , the set of natural numbers. From Galileo's paradox , there exists a bijection that maps every natural number n to its square n 2 . Since the set of squares is a proper subset of N {\displaystyle \mathbb {N} } , N {\displaystyle \mathbb {N} } is Dedekind-infinite.