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The opposite is non-strict, which is often understood to be the case but can be put explicitly for clarity. In some contexts, the word "proper" can also be used as a mathematical synonym for "strict".
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. The relationship of one set being a subset of another is called inclusion (or sometimes containment).
Standard examples of posets arising in mathematics include: The real numbers, ... By definition, every strict weak order is a strict partial order.
If the inequality is strict (a < b, a > b) and the function is strictly monotonic, then the inequality remains strict. If only one of these conditions is strict, then the resultant inequality is non-strict. In fact, the rules for additive and multiplicative inverses are both examples of applying a strictly monotonically decreasing function.
A strict total order on a set is a strict partial order on in which any two distinct elements are comparable. That is, a strict total order is a binary relation < {\displaystyle <} on some set X {\displaystyle X} , which satisfies the following for all a , b {\displaystyle a,b} and c {\displaystyle c} in X {\displaystyle X} :
The definitions can be generalized to functions and even to sets of functions. Given a function f with domain D and a preordered set (K, ≤) as codomain, an element y of K is an upper bound of f if y ≥ f (x) for each x in D. The upper bound is called sharp if equality holds for at least one value of x. It indicates that the constraint is ...
For this reason, the term strict preorder is sometimes used for a strict partial order. That is, this is a binary relation < {\displaystyle \,<\,} on P {\displaystyle P} that satisfies: Irreflexivity or anti-reflexivity: not a < a {\displaystyle a<a} for all a ∈ P ; {\displaystyle a\in P;} that is, a < a {\displaystyle \,a<a} is false for all ...
Notice that this definition approaches the definition for strict convexity as , and is identical to the definition of a convex function when = Despite this, functions exist that are strictly convex but are not strongly convex for any m > 0 {\displaystyle m>0} (see example below).