Inequation
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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
- x ≠ y.
(In programming languages and electronic communications, the notations 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 
, then x < y or x > y by the trichotomy law.
[edit] Properties
Some useful properties of inequations in algebra are:
- Any quantity can be added to both sides.
- Any quantity can be subtracted from both sides.
- Both sides can be multiplied by any nonzero quantity.
- Both sides can be divided by any nonzero quantity.
- 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.
[edit] See also
