Strict

"Strictly" redirects here. For the British reality show, see Strictly Come Dancing. For other uses, see Strict (disambiguation).

In mathematical writing, the adjective strict is used to modify technical terms which have multiple meanings. It indicates that the exclusive meaning of the term is to be understood. (More formally, one could say that this is the meaning which implies the other meanings.) The opposite is non-strict. This is often implicit but can be put explicitly for clarity. In some contexts the word "proper" is used as a mathematical synonym for "strict".

Use

This term is commonly used in the context of inequalities — the phrase "strictly less than" means "less than and not equal to" (likewise "strictly greater than" means "greater than and not equal to"). More generally a strict partial order, strict total order and strict weak order exclude equality and equivalence

A related use occurs when comparing numbers to zero — "strictly positive" and "strictly negative" mean "positive and not equal to zero" and "negative and not equal to zero", respectively. Also, in the context of functions, the adverb "strictly" is used to modify the terms "monotonic", "increasing", and "decreasing".

On the other hand, sometimes one wants to specify the inclusive meanings of terms. In the context of comparisons, one can use the phrases "non-negative", "non-positive", "non-increasing", and "non-decreasing" to make it clear that the inclusive sense of the terms is intended.

Using such terminology helps avoid possible ambiguity and confusion. For instance, upon reading the phrase "x is positive", it is not immediately clear whether x = 0 is possible, since some authors might use the term positive loosely, and mean that x is not less than zero. Therefore, it is prudent to write "x is strictly positive" for x > 0 and "x is non-negative" for x ≥ 0. (A precise term like non-negative is never used with the word negative in the wide, loose sense that includes zero.)

The word "proper" is often used in the same way as "strict." For example, a "proper subset" of a set S is a subset that is not equal to S itself, and a "proper class" is a class which is not also a set.

This article incorporates material from strict on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.