"belongs to" versus "contained in" 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? 

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.
