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

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

Andrés E. Caicedo
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mavavilj
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  • Usually, when comparing cardinalities of two sets, we write it as $|A|\leq |B|$, where $|A|$ means "the cardinality of $A$." – Thomas Andrews Sep 16 '15 at 12:53
  • @ThomasAndrews Can you refer to cardinalities without referring to functions? I've not read about cardinalities, but viewing the wikipedia page says it's defined using some injective functions. – mavavilj Sep 16 '15 at 12:54
  • Well, it depends on what you need. That's how cardinality is compared in set theory, yes, and is quite useful in comparing the sizes of infinite sets. – Thomas Andrews Sep 16 '15 at 12:57
  • You should probably avoid using them directly for that. You have the cardinality notation $|A| < |B|$. While $A<B$ might seem a good idea it would backfire if you use $A=B$ instead of $|A|=|B|$ because the former means that they have the very same elements and not just equal number of elements. In addition the $A<B$ might be interpreted as $a<b$ for all $a\in A$ and all $b \in B$. – skyking Sep 16 '15 at 12:58

3 Answers3

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$=$ 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.

Ove Ahlman
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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.

Siminore
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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|)

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 ⊊. –

Gabor Bakos
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    Note that some people use the symbol $\subset$ to mean what you've written as $\subseteq$ (despite the fact that the analogy with $<$ stops working). To be completely unambiguous, use $\subseteq$ and $\subsetneq$. – Akiva Weinberger Sep 16 '15 at 13:34
  • Is $A \subseteq B$ really always the same as $|A| ≤ |B|$? No. Let A={1,2,1}, B={1,2}. $A \subseteq B $, but $|A| ≥ |B|$. – mavavilj Sep 16 '15 at 13:47
  • A = {1,2,1} is not a valid set by definition (need to have unique elements) or is equivalent to B. A way that A might have 1,2,1 as elects is A= {{1,2},{1,}} and then A⊇B would be the case. – Gabor Bakos Sep 16 '15 at 14:16
  • @GaborBakos Good point and a reason to probably read the theory body of set theory, in order to understand the notation logic better as well. – mavavilj Sep 17 '15 at 16:36