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Produced by Charles Wells     Revised 2017-03-03

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# SUBSETS AND INCLUSION

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 of inclusion

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.

###### Examples
• $\mathbb{Q}\subseteq \mathbb{R}$ 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 $\left[ 0,\,1 \right]\subseteq \mathbb{R}$.

## Notation and terminology

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.

### Warning

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.

## Proper inclusion

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.

## Assertions form subsets

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 $\left\{ x\,|\,x\in S\text{ and }P(x) \right\}$.

##### Examples
• 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, $\left\{ z|{{z}^{2}}=-1 \right\}=\left\{ i,-i \right\}$.