# Shouldn't we refer to an inequality as an 'inequality' instead of an 'equation'?

As far as I'm concerned,

• an equation is any mathematical statement of the form LHS=RHS,
• an inequality is any mathematical statement of the form LHS?RHS (where ? is a placeholder for some inequality sign),
• an expression is any mathematical quantity (it contains no equal sign or inequality sign).

LaTeX's built-in cross-referencing to maths stuff can of course be customised with dedicated packages (such as cleveref), but it seems to me that very few TeX users actually choose to reflect that distinction in their documents; most seem quite happy to use the name Equation throughout. That bothers me a bit...

Do you think that one should use different cross-referencing names for equations, inequalities, and expressions?

Besides, what about the singular form incorrectly used instead of the plural one? Here is a baby example:

$a \leq x \leq b$

Here, we actually have, not one, but two inequalities. Isn't it wrong/confusing to refer to it as a single inequality (not to mention a single equation)?

Note: I'm conscious these questions may seem primarily opinion-based, but I feel that a limited number of those is acceptable on the site. After all, I'm sure esteemed members of the community would agree that promoting good composition style is at least part of the site's raison d'être. Feel free to disagree and migrate the question to meta or a community poll.

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Equalities (and inequalities) are expressions too. $1 = 0$ is an expression with the value "false" (usually). – Daniel Fischer Sep 30 '13 at 15:02
The example you gave has three inequalities (not just two): $a\le x$, $x\le b$, and $a\le b$. – Joel Reyes Noche Sep 30 '13 at 15:03
@JoelReyesNoche ... but the last one follows from the first two... Following your reasoning, I could say that my example contains an infinite number of inequalities... – Jubobs Sep 30 '13 at 15:03
I think people have no trouble distinguishing between the mathematical meaning of "equation" (a statement asserting that two things are equal), and the typographical meaning (a block of mathematical notation, usually set off from surrounding prose text). – Nate Eldredge Sep 30 '13 at 15:12
@DanielFischer Fair enough, an equation can be considered a logical expression, but the reverse is not true. – Jubobs Sep 30 '13 at 15:35

## 1 Answer

I agree with you. It grates on me when writers refer to an inequality as an "equation". This usage perhaps was picked up from typesetters, who were less concerned with the mathematical meaning of terms. Now that mathematicians are doing their own typesetting, one might hope that a more precise use of language will prevail. The term "inequality" does not imply a situation where just two things are unequal; for example, social inequality can refer to a society with more than two members. Likewise $$x_i\neq x_j\quad(i\neq j;\, i,\!j=1,...,n)$$ may be referred to as an inequality, although it comprises $\frac12 \!n(n-1)$ inequalities. I would also not fret about calling $a\leqslant b$ an inequality, even though it admits the possibility that $a$ and $b$ may be equal.

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