If every element of set $A$ is an element of set $B$ then we say that $A$ is subset of $B$. $A$ is said to be proper subset if $A$ does not equal $B$. My question is - whether the symbol we use to show subset is same as that for proper subset? I have seen that $A$ is subset of $A$ itself is shown by the same symbol as that for the subset while when we write improper subset, we usually write equal to sign with symbol of subset. So, please help me to know this.
I'm afraid the usage varies by author.
Some of the symbols are unambiguous: '$\subsetneq$' always means a proper subset, and '$\subseteq$' means a subset which could either be proper or not.
But the symbol '$\subset$' is ambiguous. For some, it always means a proper subset (identical to '$\subsetneq$'. For others, it just means a subset without regard to properness (identical to '$\subseteq$').
My personal preference is to use '$\subset$' to just mean any subset, proper or not. When I specifically want to exclude the possibility of equality, I explicity indicate this using '$\subsetneq$'.