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A universe set is an absorbing element of binary union . The empty set ∅ {\displaystyle \varnothing } is an absorbing element of binary intersection ∩ {\displaystyle \cap } and binary Cartesian product × , {\displaystyle \times ,} and it is also a left absorbing element of set subtraction ∖ : {\displaystyle \,\setminus :}
More generally, given a binary relation R between two sets X and Y, let E be a subset of X such that, for every , there is some such that x R y. If one has a criterion allowing selecting such a y for every x ∈ E , {\displaystyle x\in E,} this defines a function f : E → Y , {\displaystyle f:E\to Y,} called an implicit function , because it ...
When the direction of a Euclidean vector is represented by an angle , this is the angle determined by the free vector (starting at the origin) and the positive -unit vector. The same concept may also be applied to lines in a Euclidean space, where the angle is that determined by a parallel to the given line through the origin and the positive x ...
A binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is a partial order. A group with a compatible total order is a totally ordered group. There are only a few nontrivial structures that are (interdefinable as) reducts of a total order. Forgetting the orientation results in a betweenness relation.
A relation algebra (L, ∧, ∨, −, 0, 1, •, I, ˘) is an algebraic structure equipped with the Boolean operations of conjunction x∧y, disjunction x∨y, and negation x −, the Boolean constants 0 and 1, the relational operations of composition x•y and converse x˘, and the relational constant I, such that these operations and constants satisfy certain equations constituting an ...
Here the order relation on the elements of is inherited from ; for this reason, reflexivity and transitivity need not be required explicitly. A directed subset of a poset is not required to be downward closed; a subset of a poset is directed if and only if its downward closure is an ideal. While the definition of a directed set is for an ...
Every binary relation on a set can be extended to a preorder on by taking the transitive closure and reflexive closure, + =. The transitive closure indicates path connection in R : x R + y {\displaystyle R:xR^{+}y} if and only if there is an R {\displaystyle R} - path from x {\displaystyle x} to y . {\displaystyle y.}
Let F be a free ring (that is, free algebra over the integers) with the set X of symbols, that is, F consists of polynomials with integral coefficients in noncommuting variables that are elements of X. A free ring satisfies the universal property: any function from the set X to a ring R factors through F so that F → R is the unique ring ...
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