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I'm tutoring a 8th grader. Once that kid asked me if there is a unique name for $ \leq $ and $ \geq $.

Question goes like this: "Since it holds both equality $ = $ and inequality $<, >$ why is it still named inequality?"

How do I answer it?

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    It does not hold both equality and inequality at the same time.2012-08-24
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    I'd call $a \leqslant b$ just *inequality*, and $a < b$ *strict inequality*.2012-08-24
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    It is an inequality sign in the sense that it is different from the equality sign.2012-08-24
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    i see ... thanks!!2012-08-24
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    @Rahul: Yes. Probably the "true" inequality would mean $\neq$.2012-08-24
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    "Since it holds both equality ... and inequality..." - the way it's read in English already gives a strong hint that this isn't the case: "greater than **or** equal to", "less than **or** equal to". Your kid is mixing up `AND` and `OR`.2012-08-24

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The detail of $\leq$ is that it means $<$ or $=$. You might help him/her note that that $$1\leq 1$$ is a true statement, because $1$ is in fact equal or grater than $1$. It seems strange for many students to write $1\leq 1$ when it seems $1=1$ is "more true" or "better" than the former. The "problem" is that order relations (see below) are in fact defined and are analogous to the behaviour of $\leq$ and not $<$. As Halmos puts it:

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This part is for you:

In general, given a relation on a set $S$, we call it a (partial) order, and use the symbol $\leq$ or any similar variant such as $\leqslant$, $\preceq$ if it has the following properties:

NOTE: If $x\leq y$ we usually may say that $x$ is smaller or precedes $y$.

$(1)$ Transitivity If $x\leq y$ and $y \leq z$ then $x\leq z$. In words: "If a number $n$ is smaller or equal than another number $m$, and this last one is smaller or equal than another number $p$ , then first number is smaller or equal than the last one."

$(2)$ Antisymmetry If $x\leq y$ and $y\leq x$ then $y=x$. In easy words: "If a number $n$ is smaller or equal than another number $m$, and this number $m$ is smaller or equal than the first number $n$, then they must be equal. This is maybe a picky thing to explain since it has the weight of the logical operator "or": if we say $A$ or $B$ is true, then it can happen $A$ is true, $B$ is true, or both are true.

$(3)$ Reflexivity For any $x$, $x\leq x $ is always true.

Remember $\leq$ means "smaller or equal". The statement $1\leq 2$ is perfectly valid. So is $1\leq 1$. Students usually find this odd, and say "But $1$ is not smaller than $1$: true, but $\leq$ means smaller or equal. Since equality is true, the statement in question is true.

You can check that given a set $X$, usual improper set inclusion is an order in the powerset $2^X$. The reason it is called a partial order is that sometimes we cannot relate two elements. For example, ${1,2}$ and ${3,4}$ are both in $2^{\{1,2,3,4\}}$, but neither $\{1,2\} \subseteq \{3,4\}$ nor $\{3,4\} \subseteq \{1,2\}$ hold.

An order is called a total order if for any $x,y$ in the set in question, either $x\geq y$ or $y\geq x$ holds - intuitively, we can compare every pair of elements. The usual inequality of numbers is a strict order.

Now, associated to each (partial) order $\leq$ is the relation $<$: we say that $x iff $x\leq y$ and $x\neq y$. This new order relation arising from the old one is transitive, and for no elements $x,y$ do both $x and $y hold simultanesouly. This is usually called the strict order relation corresponding to $\leq$.

All that technicality is intended for you to have a good idea about what order relations are in general. Hope it helps,

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    A Nice one. : )2012-08-24
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    ...nice segue into posets. :)2012-08-24
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I'd call them "non-strict inequalities".

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They are "less-than-or-equal-to" and "greater-than-or-equal-to", but since mathematicians are lazy, they write $\le$ and $\ge$. If they have to use the (English) alphabet, they use "le" and "ge", as in fortran and TeX (with appropriate bracketing).

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    When did *elegance* became *lazy*!2012-08-24
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    Mathematicians are notoriously lazy, and that's a good thing (it leads to discovering better ways of doing things). But in matters of notation, clarity rather than laziness is the main concern. For example, we don't generally use Polish notation, even though it would save a few parentheses, because it's harder for humans to read.2012-08-24
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    @Jen, you haven't heard of "enlightened laziness" before? :)2012-08-24
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I'd call them ``slack inequalities".

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    Why "slack"? Is this serious?2012-08-24
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    @Peter Well, it's not! But, I am habituated to calling inequalities strict and slack.2012-08-24
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    Slack as in optimization's [slack variables](http://en.wikipedia.org/wiki/Slack_variable)?2012-08-24
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    @JenniferDylan Yes, I had this in mind. :-)2012-08-24