2.0: Introduction
- Page ID
- 10064
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Sets
A set is a collection of objects. These objects are called elements (or members) of the set. Sets are normally denoted by capital letters (\(A, B, C, \dots\)), and their elements by lowercase letters (\(a, b, c, \dots\)).
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 elements.
Example \(\PageIndex{1}\)
Let \(A = \{1, 2, 3, 4, 5\}\),
Then
- \(1 \in A\) (\(1\) is an element of (or belongs to) set \(A\))
- \(0 \notin A\) (\(0\) is not an element of (or does not belong to) set \(A\)).
Set Builder Notation
"Set builder notation" is used to express sets in which a pattern is present. That is, set-builder notation is used to specify a set by stating the properties that its elements must satisfy.
Let \(C\) be the set of all positive integers. Using set-builder notation:
\( C = \{ x \in \mathbb{Z} \mid x > 0 \} \)
This is read as: ``\(C\) is the set of all \(x\) in \(\mathbb{Z}\) such that \(x\) is greater than zero.''
Example \(\PageIndex{3}\)
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?
We can express \(D\) in set-builder notation as:
\(D = \{ x \in \mathbb{Z} \mid x > 0 \text{ and } x \text{ is odd}\} = \{2k - 1 \mid k \in \mathbb{Z}^+\}\)
Example \(\PageIndex{3}\)
Consider \(\mathbb{Q}\), the set of rational numbers. Rational numbers are numbers that can be expressed as fractions \(\frac{m}{n}\) where \(m, n \in \mathbb{Z}\) and \(n \neq 0\):
\(\mathbb{Q} = \left\{ \frac{m}{n} \;\middle\vert{}\; m, n \in \mathbb{Z}, n \neq 0 \right\}\)
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 denotes the imaginary unit.
- \(\mathbb{Q}\) represents the set of all rational numbers: \(\mathbb{Q} = \left\{\frac{m}{n} \mid m, \, n \in \mathbb{Z}, \, n \neq 0 \right\}.\)
- \(\mathbb{Q}\)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 of which all others are subsets.