2.0: Introduction

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Sets

A set is a collection of things. These things are called elements of the set. Sets are normally denoted by using capital letters, and the elements are denoted using small letters. We write \(a \in A\) for "a is an element of a set A", and \(a \notin A\), for "a is not an element of a set A". \(\emptyset\) or \(\{\}\) denotes the empty set, which contains no element.

Example \(\PageIndex{1}\)

Let \(A = \{1, 2, 3, 4, 5\}\),

Then 1 is an element of (or belongs to) set A, we write:

\(1 \in A\)

and 0 is not an element of A, we write:

\(0 \notin A\).

Set Builder Notation

"Set builder notation" is used to express sets in which a pattern is present. Consider if set C is the set of all positive integers. Instead of writing down each one, how could we express set C in a general form?

This set would be written:

\(C = \{x \in \mathbb{Z} \mid 0 < x \}\). This would read "Set C contains integers x, where x is greater than zero."

Example \(\PageIndex{2}\)

Consider set \(D = \{1, 3, 5, 7...\}\):

\(D\) consists of positive, odd integers, or \(x \in \mathbb{Z}, \, x > 0, \, 2 \nmid x\).

So:

\(D = \{x \in \mathbb{Z} \mid x > 0, \, 2 \nmid x \}\).

Could we use any other sets to define \(x\)? Which ones would work? Which ones would not?

Example \(\PageIndex{3}\)

Consider \(\mathbb{Q}\), the set of rational numbers. How might we express \(\mathbb{Q}\) in set builder notation?

Rational numbers are expressed as repeating or terminating numbers, which can be expressed as fractions: \(\frac{m}{n}\).

In fractions, the denominator must not be zero: \(n \neq 0\).

Also, fractions cannot have decimals as terms, so \(m\) and \(n\) must be \(\in \mathbb{Z}\).

Instead of integers, if we used whole or natural numbers, we would miss out on the negative values. Thus, \(m, \, n \in \mathbb{Z}\).

So:

\(\mathbb{Q} = \{\frac{m}{n} \mid m, \, n \in \mathbb{Z}, \, n \neq 0 \}\)

We can see, using set builder notation, that any number capable of being expressed as a fraction \(\in \mathbb{Q}\).

Notations:

We can use set notation to specify and help describe our standard number systems. The following standard sets are given from smallest to biggest:

\(\mathbb{N}\) represents the set of all natural numbers: \(\mathbb{N} = \{1, 2, 3, 4...\}\)
\(\mathbb{W}\) represents the set of all whole numbers: \( \mathbb{W} = \{0, 1, 2, 3...\} \)
\(\mathbb{Z}\) represents the set of all integers: \(\mathbb{Z} = \{ ...-2, -1, 0, 1, 2...\}\). \(i\) is not used because it is used for complex numbers.
\(\mathbb{Q}\) represents the set of all rational numbers: \(\mathbb{Q} = \{0, \pm1, \pm\frac{1}{2}, \pm\frac{1}{3}...\}\)
\(\mathbb{Q}\)^c represents the set of all irrational numbers
\(\mathbb{R}\) represents the set of all real numbers
\(\mathbb{U}\) represents the universal set, the set to which all others are a subset.