2. Describing sets by properties

Describing sets by properties

Finite set

A finite set is a set that has a finite number of elements. We can informally define it as a set which one could count and finish counting.

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Infinite set

An infinite set is a set which is not a [[Finite set|finite set]], there is no definitive end.

Examples include:

• Set of all integers

  $\mathbb{Z}=\{...,-3,-2,-1,0,1,2,3,...\}$ is the set of all integers.

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• Set of natural numbers

  $\mathbb{N}=\{0,1,2,3,...\}$ is the set of all natural numbers.

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• Set of positive natural numbers

  ${\mathbb{N}}^{+}=\{1,2,3,...\}$ is the set of all positive natural numbers, or otherwise all positive integers.

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• Set of rational numbers

  $\mathbb{Q}=\{\frac{m}{n},...\}$ is the set of rational numbers, any number that can be expressed as $\frac{m}{n}$ for some integers $m$ and $n \ne 0$.

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• Set of real numbers

  $\mathbb{R}=\{-1,0,1,\pi ,\sqrt{2}...\}$ is the set of real numbers, which includes all rational and irrational numbers.

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• Set of positive real numbers

  ${\mathbb{R}}^{+}=\{1,2,\pi ,...\}$ is the set of positive real numbers.

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Set builder notation

Describing a set by listing is not ideal, we can instead describe a property that the elements of the set satisfy.

If $\mathcal{P}$ is a property, then the set $S$ whose elements hold this property is denoted by:

$$\begin{array}{rl}S& =\{x|x\text{ has property }\mathcal{P}\}\\ & =\{x|P(x)\}\end{array}$$

$S$ is the set of all $x$ such that $x$ has property $\mathcal{P}$.

If we know that all elements in $S$ come from a larger set $A$:

$$S=\{x\in A|x\text{ has property }\mathcal{P}\}$$

For example, let $\text{Odd}$ be the set of all odd integers, we can describe this in a number of different ways:

$$\begin{array}{rl}\text{Odd}& =\{...,-5,-3,-1,1,3,5,...\}\\ & =\{x|x\text{ is an odd integer}\}\\ & =\{x\in \mathbb{Z}|x\text{ is odd}\}\\ & =\{x|x=2k+1\text{ for some integer }k\}\\ & =\{x|x=2k+1\text{ for some }k\in \mathbb{Z}\}\\ & =\{2k+1|k\text{ is an integer}\}\\ & =\{2k+1|k\in \mathbb{Z}\}\end{array}$$