Give the definition of S $\subseteq$ T for general sets S and T.

Question

The answer I can come up with is;

S is a subset of T, denoted by S $\subseteq$ T, or equivalently, T is a superset of S, denoted by T $\supseteq$ S.

Can someone correct me if I am wrong, or provide a better answer? Thank you.

Answer 1

Your answer is not really good (as far as I can guess), because $S\subseteq T$ is pronounced "$S$ [is a] subset [of] $T$", so you essentially define subset by saying that it means being a subset.

My guess is that the question is asking you give give the mathematical definition of being a subset, which I'll leave for you to locate in your notebook.

Answer 2

We say that $S$ is a subset of $T$ if any element of $S$ is an element of $T$, and we say that $S$ is a proper subset of $T$ if $S$ is a subset of $T$ and $S\neq T$. Being a [proper] superset is the opposite relation.

$\subseteq$ is a symbol denoting the relation of inclusion (i.e. of being a subset). Nowadays it has (from what I've seen) nearly completely superseded $\subset$, having the advantage of being less ambiguous: some authors would use $\subset$ to denote proper inclusion (like we differentiate between $<$ and $\le$).

A modern, unambiguous convention is to write $\subsetneq$ to denote proper inclusion, and $\subseteq$ to denote any (possibly improper) inclusion. Both have symmetric counterparts, namely $\supsetneq,\supseteq$.