Search results
Results from the WOW.Com Content Network
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 :}
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.
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.}
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 ...
(x 1, y 1) + (x 2, y 2) = (x 1 + x 2, y 1 + y 2). Let R + be the group of positive real numbers under multiplication. Then the direct product R + × R + is the group of all vectors in the first quadrant under the operation of component-wise multiplication (x 1, y 1) × (x 2, y 2) = (x 1 × x 2, y 1 × y 2). Let G and H be cyclic groups with two ...
If f is an n-ary function symbol, and t 1, ..., t n are terms, then f(t 1,...,t n) is a term. In particular, symbols denoting individual constants are nullary function symbols, and thus are terms. Only expressions which can be obtained by finitely many applications of rules 1 and 2 are terms.
A representation of an ordering on a set of vectors is a function f that assigns a single number to each vector, such that the ordering between the numbers is identical to the ordering between the vectors. That is, f(x) ≥ f(y) iff x is larger than y by that ordering. When the number of possible vectors is countable (e.g. when all vectors are ...
Binary trees may also be studied with all nodes unlabeled, or with labels that are not given in sorted order. For instance, the Cartesian tree data structure uses labeled binary trees that are not necessarily binary search trees. [4] A random binary tree is a random tree drawn from a certain probability distribution on binary trees. In many ...