Let us consider a set $A$. let $B$ be an element of the set. Now what I want to know is that whether saying $B$ is contained in $A$ and $B$ belongs to $A$ means the same? Could anyone here cite any context where they do not mean the same?

  • $\begingroup$ user16186, Can you clarify this one point? Is your confusion about how the two relations $a \in A$ and $B \subseteq A$ are different? Or are you asking about the terminology? $\endgroup$ – Srivatsan Sep 16 '11 at 17:57

Paul Halmos in his autobiography reports that he once decided that henceforward he would say "$x$ contains $y$" when he meant "$y$ is a member of $x$" and "$x$ includes $y$" when he meant "$y$ is a subset of $x$". He adhered to that usage fastidiously for 18 months. At the end of that time he drew his conclusions: (1) the practice is harmless, and (2) he didn't think anybody ever noticed.

I was inclined to agree with the usage on the grounds that people speak of a family of subsets being "partially ordered by inclusion" but they never say "partially ordered by containment" as far as I know.

And as far as I know, "$x$ belongs to $y$" would meant the same thing as "$x$ is a member of $y$".

But sometimes people say "$x$ is contained in $y$" when they mean $x$ is a subset of $y$. And sometimes they say the same thing when they mean $x$ is a member of $y$. So always make it clear which meaning you have in mind. Sometimes context is enough for that and probably sometimes it is not.

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  • $\begingroup$ For example, here's a sentence taken from wikipedia: Indeed, a set is closed if and only if it contains all of its limit points. It uses "$X$ contains $x$" to mean $x \in X$. $\endgroup$ – Srivatsan Sep 16 '11 at 17:50

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