Comparison (<, >, ...) notation for sets

Question

Are comparison notations such as <, >, ≤, ≥, =, ≠ valid for sets?

I'm interested in stating size (number of elements) relations between sets.

Answer 1

Equality of course is. I have never seen the notation $A<B$ to say that $A$ has less elements than $B$. Everybody uses a more descriptive notation as $\#A < \# B$, for instance.

Answer 2

$=$ and $\neq$ are defined and valid (see the axiom of extensionality).

$\leq, \geq, <$ and $>$ are more tricky. We may compare the number of elements in two sets, called the cardinality, and write that $|A|> |B|$ meaning that the cardinality of $A$ is greater than B, however this notation is never used directly on sets i.e. without the use of bars around the sets .

On the other hand $\subseteq, \subset,\supseteq $ and $\supset$ (the subset and superset relation) works just like ordering on sets, so one might argue that writing $A<B$ means $A\subseteq B$ as it does so in the subset partial order which is induced between all sets.

In general however $\leq, \geq, <$ and $>$ are not defined explicitly for sets, and If you're reading litterature which uses this notation, you need to look up what they define it as.

Answer 3

You can use the $=$ sign, which has the definition: both sets have the same members.

The other comparison notations are not used unless you are comparing the sizes (cardinality, the number of elements) of the sets:

A={1,2,3,4}, B={1,5}, here $≥$ can be used (|A|≥|B|)

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Some comparison notations which look similar but mean different things:

$⊆$ (similar to $≤$): subset has fewer elements or equal to the set

{9,14,28} $⊆$ {9,14,28}

$⊇$ (similar to $≥$): set A has more elements or equal to the set B

{9,14,28} $⊇$ {9,14,28}

$⊊$ (similar to $<$): subset has fewer elements than the set

{9,14} $⊂$ {9,14,28}

$\supsetneq$ (similar to $>$): set A has more elements than set B

{9,14,28} $⊃$ {9,14}

Note that some people use the symbol ⊂ to mean what you've written as ⊆ (despite the fact that the analogy with < stops working). To be completely unambiguous, use ⊆ and ⊊. –