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
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.
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.
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
See also
{{Wiktionary}}
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.
لامعادلةInéquationDisequazioneНееднаквост (математика)Inequação不等式
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