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What is the difference between saying that something is a subset of and something is contained in? I am studying basic set theory on my own and this is one of the finer points I feel is important but am unable to grasp.

Prakhar Nagpal
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    Does https://math.stackexchange.com/questions/131309/set-theory-difference-between-belong-contained-and-includes-subset help? – BallBoy Jan 09 '18 at 02:06
  • A set contains elements and one way of writing a (finite) set is by writing its elements enclosed in a pair of brackets. For example the set ${3,5,8}$ is a three-element set which contains the elements $3,5$ and $8$. To notate that $3$ is an element of ${3,5,8}$ we write $3\in {3,5,8}$. Meanwhile, a subset $A$ of another set $B$ is a set where every element in $A$ (if any exist) is also an element of $B$, for example ${3,5}$ is a subset of ${3,5,8}$. To notate this, we write ${3,5}\subset{3,5,8}$. – JMoravitz Jan 09 '18 at 02:09
  • In this example $3$ is an element of ${3,5,8}$, $3$ is not a subset of ${3,5,8}$ because $3$ is not a set, ${3}$ is not an element of ${3,5,8}$ because the set ${3}$ is not an element included within the brackets (Compare to the set ${{3},5,8}$ where ${3}$ would have been considered an element) (this is despite the element IN ${3}$ being an element in ${3,5,8}$) and finally ${3}$ is a subset of ${3,5,8}$ because every element in ${3}$ (namely $3$) is also an element of ${3,5,8}$. – JMoravitz Jan 09 '18 at 02:12

2 Answers2

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That a set is contained another set usually means
it is a subset. That an element is contained
in a set means it is in the set or a member of
the set. Much to be prefered is the expression
"is in". To be avoided is using in to mean subset.

William Elliot
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In modern mathematics, the English language is only used to make our (math) statements in a more comfortable way. But it can always be made more formal with symbols, signs, operators, relations, etc.

So, if you feeling confused about English statements, see if you can transcribe them into more formal expressions.

So if you see "belongs to", "is a subset of", "is contained in" in most contexts you have to set up formal statements using either $\in$ or $\subset$. Also, you might at times be looking at $\subseteq$, $\subsetneq$, $\supset$, or $\notin$.

Now, to be honest, I really don't know what is the difference between $\subset$ and $\subseteq$. But if I am reading something and the author likes to examine proper subsets, then I will adjust accordingly. If you are reading different books on set theory you might have to get a feel for 'different tempos' and blend it all together into understandable chunks.

CopyPasteIt
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