a set A
is a subset
of a set B
if the following material implication is true:

Consequently, the basic idea is that all of the elements of A are elements of B.

A "horse shoe" symbol on its side is used to represent the subset relationship between sets.
Thus, means that A is a subset of B.

The horse shoe can go either way, so that also means that A is a subset of B.
However, this second way of writing the relation is usually read as "B contains A".

2 ---------- begin topic:

The Null Set as a Subset

We can make sense of the confusing but important fact
that the null set is a subset of any set, A, by looking at the following material implication:

This conditional is true because, no matter what x is, its premise is always false,
and a conditional with a false premise is always true:
See the truth table.

Thus, by the definition given above, we have for any set A.

3 ---------- begin topic:

Any Set as a Subset

It is also true that any set is a subset of itself.
In this case the definition of subset gives us for any set A


Now both the premise and the conclusion are either both true or both false.
So, by the truth table, this if-then statement is always true, and A is a subset of itself.

This relation is called improper.

If the "larger" set has elements that are not in the "smaller" one, then the subset relation is called proper.