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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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    "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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    ...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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    @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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    @JenniferDylan Yes, $I$ had this in mind. :-)2012-08-24