Produced by Charles Wells Revised 2017-03-03 Introduction to this website website TOC website index blog Back to head of Sets chapter

Every integer is a rational number.
This means that the sets
$\mathbb{Z}$ and $\mathbb{Q}$ have a special relationship to each other: *every
element of $\mathbb{Z}$ is an element of $\mathbb{Q}$*. This is the relationship
captured by the following definition.

Definition: For all sets $A$ and $B$, the assertion "$A\subseteq B$" is true if and only if every element of $A$ is also an element of $B$.

This is a mathematical definition. It means that **both the following statements are true** for all sets $A$ and $B$:

If you know that $A\subseteq B$,

then you know that every element of $A$ is an element of $B$.

If you know that every element of $A$ is an element of $B$,

then you know that $A\subseteq B$.

The definition of inclusion gives a **rule of inference**, described in the article Set: Rules of Inference.

- $\mathbb{Q}\subseteq \mathbb{R}$ and $\mathbb{Z}\subseteq \mathbb{R}$.
- $\left\{ 1,\,2,\,3 \right\}\subseteq \left\{ 1,\,2,\,3,\,4 \right\}$.
- The interval $\left[ 0,\,1 \right]\subseteq\left[ 0,\,2\right]$.
- The interval $\left[ 0,\,1 \right]\subseteq \mathbb{R}$.

The statement "$A\subseteq B$" is read as:

- "$A$ is
**contained in**$B$", - "$A$ is
**included in**$B$", or - "$A$ is a
**subset of**$B$."

All three of those statements mean exactly the same thing.

**Fact:** For every set $A$, $A\subseteq A$. In
other words, every set is a subset of itself.

**Fact:** If $A\subseteq B$ and $B\subseteq A$ then $A=B$. In other words, if $A$ and $B$ include each other, then they are the same set.

**Fact:** For every set $A$, $\varnothing \subseteq
A$. In other words, the empty set is a subset of every set. Proof.

The notion that the empty set is an *element* of every set is a **myth** widely believed by undergraduate math students. Get over it.

The statement that every set is a subset of itself can cause cognitive dissonance, because the "sub" prefix may lead you to believe that it is saying "$A$ is smaller than itself." The phrases "contained in" and "included in" also can cause cognitive dissonance for similar reasonss.

The assertion "$A\,\subsetneq B$" ($A$ is a proper subset of $B$) means that every element of $A$ is an element of $B$, but there is at least one element of $B$ that is not an element of $A$.

For example, $\mathbb{Z}\,\,\subsetneq \,\mathbb{R}$ because every integer is a real number but there are real numbers that are not integers. (See proper for some ambiguity in the use of this word.)

The notation for inclusion has gotten **Horribly Messed Up** in the last fifty years.

The problem is that in the mathematical research literature, the expression "$A\subset B$" means $A\subseteq B$, whereas in many college texts, and invariably in high school, "$A\subset B$" means $A\subsetneq B$. The sad story is discussed in detail in the section on symbols.

Because of this confusion, I do not use the expression "$A\subset B$" in abstractmath.org except when I am talking about this major notational disaster.

If $P(x)$ is an assertion whose only variable is $x$ then the set of elements of a set $S$ for which $P(x)$ is true is a subset of $S$. Using setbuilder notation, this subset is denoted by $\left\{ x\,|\,x\in S\text{ and }P(x) \right\}$.

- Let $S$= $\{2, 3, 4, 5 ,6\}$ and let $P(n)$ be the assertion "$n$ is an even integer". Then $P$ determines the subset $\{2, 4, 6\}$ of $S$, so we could write \[\{2, 4,6\}=\left\{ n\,|\,n\in S\text{ and }n\text{ is even}\right\}\].
- The circle of radius $1$ with center at the origin is the subset $\left\{ (x,y)\,|\,{{x}^{2}}+{{y}^{2}}=1 \right\}$ of the $xy$ plane.
- If $x$ is a real variable, then $\left\{ x\,|\,{{x}^{2}}=-1 \right\}$ is a subset of the set $\mathbb{R}$ of real numbers. It is the empty set.
- If $z$ is a complex variable, then the set $\left\{ z\,|\,{{z}^{2}}=-1 \right\}$ is a subset of $\mathbb{C}$. In fact, $\left\{ z|{{z}^{2}}=-1 \right\}=\left\{ i,-i \right\}$.

This work is licensed under a Creative Commons Attribution-ShareAlike 2.5 License.