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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.
This exact hitting set example is essentially the same as the detailed example above. Displaying the relation is contained in (∈) from elements to subsets makes clear that we have simply replaced lettered subsets with elements and numbered elements with subsets: a ∈ I, IV, VII; b ∈ I, IV; c ∈ IV, V, VII; d ∈ III, V, VI; e ∈ II, III ...
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 ...
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
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 ...
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.
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.
A subset of a poset is downward-directed if and only if its upper closure is a filter. Directed subsets are used in domain theory, which studies directed-complete partial orders. [5] These are posets in which every upward-directed set is required to have a least upper bound. In this context, directed subsets again provide a generalization of ...