Inequation
From Wikipedia, the free encyclopedia
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
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.
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[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] Systems of inequations
A system of inequations can be represented by a set of n variables {x1, x2, … xn} and a set of inequations involving some (possibly empty) subset of all pairs of variables (xi, xj) 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
[edit] See also
[edit] References
- Duffany, J.L., "Systems of Inequations", Fourth LACCEI International Latin American and Caribbean Conference on Engineering and Technology (LACCEI ‘2006) “Breaking Frontiers and Barriers in Engineering: Education, Research and Practice”, 21–23 June 2006, Mayaguez, Puerto Rico.