Inequation

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

In programming languages and electronic communications, the notations

x != y

x /= y

x <> y

and others, are used instead.

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 x ≠ y then x < y or x > y by the trichotomy law.

Verbally it may be spoken as "does not equal." For example the written inequation "3 ≠ 2" would be spoken as "Three does not equal two."

Properties

Some useful properties of inequations in algebra are:

1. Any quantity can be added to both sides.
2. Any quantity can be subtracted from both sides.
3. Both sides can be multiplied by any nonzero quantity.
4. Both sides can be divided by any nonzero quantity.
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.

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.

Systems of inequations

A system of inequations can be represented by a set of n variables {x₁, x₂, … x_n} and a set of inequations involving some (possibly empty) subset of all pairs of variables (x_i, x_j) for i ≠ j. The idea is analogous to a system of equations, since any valid solution must simultaneously satisfy all of the inequations in the system. For example if n = 2 the system is represented by a single inequation

See also

Look up inequation in Wiktionary, the free dictionary.

• Equation
• Inequality
• Relational operator

References

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