Inequation: Difference between revisions
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(In [[programming language]]s and electronic communications, the notations <code>''x'' != ''y''</code> and <code>''x'' <> ''y''</code> are used instead.) |
(In [[programming language]]s and electronic communications, the notations <code>''x'' != ''y''</code> and <code>''x'' <> ''y''</code> are used instead.) |
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Inequations should not be confused with mathematical [[inequality|inequalities]], which express numerical relations such as 3 < 5 ('3 is less than 5'). |
Inequations should not be confused with mathematical [[inequality|inequalities]], which express numerical relations such as 3 < 5 ('3 is less than 5'), but notice that in an [[ordered field]] inequation is an inequality. If <math>x \neq y</math>, then <math>x < y</math> or <math>x > y</math> by the [[trichotomy law]]. |
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==Properties== |
==Properties== |
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Revision as of 22:40, 22 February 2006
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 and x <> y are used instead.)
Inequations should not be confused with mathematical inequalities, which express numerical relations such as 3 < 5 ('3 is less than 5'), but notice that in an ordered field inequation is an inequality. If , then or 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.
- Any nonzero quantity can be multiplied to both sides.
- Any nonzero quantity can divide both sides.
- 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.