##
*
Subsets
*

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*".

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##
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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*.