In mathematics, an inequation is a statement that two objects or expressions are not the same, or do not represent the same value. This relation is written with a crossed-out equal sign, like Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... An expression in the very basic sense is the noun form of the verb express. ... Look up Relation in Wiktionary, the free dictionary In mathematics, a relation is a generalization of arithmetic relations, such as = and <, which occur in statements, such as 5 < 6 or 2 + 2 = 4. See relation (mathematics), binary relation (of set theory and logic) and relational algebra. ... See also the disambiguation page title equality. ...

x ≠ y.

(In programming languages and electronic communications, the notations x != y, x <> y, and x # y are used instead.) A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ...

Inequations should not be confused with mathematical inequalities, which express numerical relations such as 3 < 5 ('3 is less than 5'). In a linearly ordered set, any inequation implies an inequality: if , then x < y or x > y by the trichotomy law. The feasible regions of linear programming are defined by a set of inequalities. ... In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... Generally, a trichotomy is a splitting into three disjoint parts. ...

Properties

Some useful properties of inequations in algebra are: Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. ...

1. Any quantity can be added to both sides.
2. Any quantity can be subtracted from both sides.
3. Any nonzero quantity can be multiplied to both sides.
4. Any nonzero quantity can divide both sides.
5. Generally, any injective function can be applied to both sides.

Property (5) is somewhat of a tautology, since injective functions may be defined as functions that always preserve inequations. 3 + 2 with apples, a popular choice in textbooks Addition is the basic operation of arithmetic. ... 5 - 2 = 3 Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. ... In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... Within the study of logic, a tautology is a statement containing more than one sub-statement, that is true regardless of the truth values of its parts. ...

If a function that is not injective is applied to both sides of an inequation, the resulting statement may be false. For an extreme example, if f is a constant function, such as multiplication by zero, then the statement "f(x)≠f(y)" is always false. This consideration explains why one must use a nonzero quantity in property (3) above. False is the antonym of the adjective true. ... In mathematics a constant function is a function whose values do not vary and thus are constant. ... 0 (zero) is both a number — or, more precisely, a numeral representing a number — and a numerical digit. ...

See also

• Equation