2 Sets and functions 2.1 Introduction to set theory 2.3 Set algebra

2.2 Set operators

2.2.1 Union, intersection, difference, complement

Definition 2.2.1.

Let $A$ and $B$ be sets.

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$A\cup B$ , the union of $A$ and $B$ , is $\{x:x\in A\vee x\in B\}$ .
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$A\cap B$ , the intersection of $A$ and $B$ , is $\{x:x\in A\wedge x\in B\}$ .
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$A\setminus B$ , the set difference of $A$ and $B$ , is $\{x\in A:x\notin B\}$ .
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If $A$ is a subset of a set $\Omega$ then $A^{c}$ , the complement of $A$ in $\Omega$ , is $\{x\in\Omega:x\notin A\}$ .

We can express complements using set differences. If $A$ is a subset of $\Omega$ then its complement $A^{c}$ in $\Omega$ is equal to $\Omega\setminus A$ .

Example 2.2.1.

Suppose $A=\{0,1,2\},B=\{1,2,3\},C=\{4\}$ .

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$A\cup B=\{0,1,2,3\}$
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$A\cap B=\{1,2\}$
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$A\cap C=\emptyset$
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$A\setminus B=\{0\}$
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$A\setminus C=A$
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If $\Omega=\{0,1,2,\ldots\}$ then $A^{c}$ would be $\{3,4,\ldots\}$ .

The set $\mathbb{N}=\{0,1,2,\ldots\}$ is called the natural numbers. Some people exclude 0 from $\mathbb{N}$ but in MATH0005 the natural numbers include 0.

It’s typical to draw Venn diagrams to represent set operations. We draw a circle, or a blob, for each set. The elements of the set $A$ are represented by the area inside the circle labelled $A$ . Here are some examples:

Figure 2.1: Venn diagram for the set difference of

A

and

B

Figure 2.2: Venn diagram for the union of

A

and

B

Figure 2.3: Venn diagram for the intersection of

A

and

B

Figure 2.4: Venn diagram for the complement of the union of

A

and

B

Figure 2.5: Venn diagram for the complement of the intersection of

A

and

B

2.2.2 Size of a set

Definition 2.2.2.

The size or cardinality of a set $X$ , written $|X|$ , is the number of distinct elements it has.

Example 2.2.2.

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$|\{1,2\}|=2$
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$|\emptyset|=0$
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$|\{1,2,1,3\}|=3$

Definition 2.2.3.

A set is finite if it has 0, or 1, or 2, or any other natural number of elements. A set that is not finite is called infinite.

$\mathbb{N}$ and $\mathbb{Z}$ are infinite sets while the sets in Example 2.2.2 are all finite.