Why should $\subset$ denote "subset"?

Question

Some mathematicians use the symbol $\subset$ to denote that a set is a subset of another set, i. e., $$A\subset B\quad:\iff\quad\text{Every element of $A$ is also an element of $B$}.$$ Thus, using this definition, $A\subset A$ for all set $A$. I find this distasteful, since $m<n$ in arithmetic means that $m$ is strictly less than $n$. If find it more intuitive to use $\subset$ for "proper subset", and $\subseteq$ for "subset".

Why do some mathematician define $A\subset B$ to mean that $A$ is a subset (not necessarily proper subset) of $B$?

Answer 1

Not speaking for other mathematicians, but in general, the notation is not as important as being explicit about what one means. Therefore, it is less helpful to say $A \subset B$ or $A \subseteq B$ than it is to say "Let $A$ be a proper subset of $B$," or what have you.

Answer 2

Notation are used for our convenience, Some mathematician may use $\subseteq$ for subset while other use $\subset$. I prefer to use $\subseteq$ for subset and $\subset$ for proper subset.we may use any notation for our convenience,but you need to define it earlier.