Beginner questions about Sets.

Question

I have a quick question in regard to sets. I am a little confused when I see the notation $A\subseteq B$. How is this different than the sets $A$ and $B$ being identical? I guess some of the confusion began when I was introduced to the idea of proper subsets as well. For a subset, does $A$ only contain elements in $B$? What is the difference between subset and equal? and what is the difference between subset, superset and proper subsets? Any good examples would be great. Thanks everyone for the help!

Answer 1

Let $A$ and $B$ be sets.

We say that $A$ is a subset of $B$, written $A \subseteq B$, if every element of $A$ is an element of $B$. For example, $\{1, 2\} \subseteq \{1, 2, 3\}$

We say $A$ is a superset of $B$, written $A \supseteq B$, if every element of $B$ is an element of $A$. For example, $\{a, b, c\} \supseteq \{a, b\}$.

Useful Exercises:

• $A \subseteq B$ if and only if $B \supseteq A$
• $A = B$ if and only if $A \subseteq B$ and $A \supseteq B$.

We say $A$ is a proper subset of $B$, written $A \subset B$, if $A$ is a subset of $B$ but $A \neq B$. For example, $\{1, 2\} \subset \{1, 2, 3\}$.

The notation is incredibly similar to the notation used with inequalities (i.e. $\leq, \geq,$ and $<$) and may be a useful analogy moving forward.

Added Later: I should point out that the use of notation I've indicated above is not entirely universal. Some people choose to use $\subset$ to mean subset rather than proper subset, but (in my experience) this is less common.

Answer 2

Two sets are equal if they have exactly the same elements, but $A\subseteq B$ if all we know is that all the elements of $A$ are elements of $B$.

For example, $\Bbb N\subseteq\Bbb Z$, but they are not equal because $-1\notin\Bbb N$.

It should be noted, though, that it is always true that $A\subseteq A$. And in fact $A=B$ if and only if $A\subseteq B$ and $B\subseteq A$.