2.0: Introduction
- Page ID
- 10064
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Which of the following describe well-defined mathematical sets?
- The months containing 31 days.
- The best movies ever made.
- The even integers.
- The most beautiful numbers.
Sets
A set is a well-defined collection of distinct objects, called its elements. That means a set is a collection where we can determine whether
a given object belongs to the collection.
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 \(A\)).
The order of elements does not matter, and repeated elements do not change a set.
For example,
\[ \{2,4,6\}=\{6,2,4,2\}. \]
Set Builder Notation
"Set builder notation" is used to express sets in which a pattern is present. That is, set-builder notation specifies a set by stating the properties its elements must satisfy.
\( A=\{x\in U:P(x)\}, \)
where \(P(x)\) is a condition that determines whether \(x\) belongs to \(A\). Note that
\( \{x\in\mathbb Z:x^2=9\}=\{-3,3\} \)
and
\( \{x\in\mathbb N:x^2=9\}=\{3\}. \)
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 \(\PageIndex{4}\)
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 \(\PageIndex{5}\)
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 \(n \neq 0\):
\(\mathbb{Q} = \left\{ \dfrac{m}{n} \;\middle\vert{}\; m, n \in \mathbb{Z}, n \neq 0 \right\}\)
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.