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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.
2. In the graphic matroid of a graph, a subset of edges is independent if the corresponding subgraph is a tree or forest. In the bicircular matroid, a subset of edges is independent if the corresponding subgraph is a pseudoforest. indifference An indifference graph is another name for a proper interval graph or unit interval graph; see proper ...
may mean that A is a subset of B, and is possibly equal to B; that is, every element of A belongs to B; expressed as a formula, ,. 2. A ⊂ B {\displaystyle A\subset B} may mean that A is a proper subset of B , that is the two sets are different, and every element of A belongs to B ; expressed as a formula, A ≠ B ∧ ∀ x , x ∈ A ⇒ x ∈ ...
The expressions "A includes x" and "A contains x" are also used to mean set membership, although some authors use them to mean instead "x is a subset of A". [2] Logician George Boolos strongly urged that "contains" be used for membership only, and "includes" for the subset relation only. [3] For the relation ∈ , the converse relation ∈ T ...
If a partially ordered set admits a totally ordered cofinal subset, then we can find a subset that is well-ordered and cofinal in . If ( A , ≤ ) {\displaystyle (A,\leq )} is a directed set and if B ⊆ A {\displaystyle B\subseteq A} is a cofinal subset of A {\displaystyle A} then ( B , ≤ ) {\displaystyle (B,\leq )} is also a directed set.
Suppose that G is a group, and H is a subset of G. For now, assume that the group operation of G is written multiplicatively, denoted by juxtaposition. Then H is a subgroup of G if and only if H is nonempty and closed under products and inverses. Closed under products means that for every a and b in H, the product ab is in H.
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A subset S of F is a base or basis for F if the upper set generated by S (i.e., the smallest upwards-closed set containing S) is equal to F. Since every filter is upwards-closed, every filter is a base for itself. Moreover, if B ⊆ P is nonempty and downward directed, then B generates an upper set F that is a filter (for which B is a base).