4.2: Sequences
- Page ID
- 48312
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Sets provide one way to group a collection of objects. Another way is in a sequence, which is a list of objects called terms or components. Short sequences are commonly described by listing the elements between parentheses; for example, (\(a, b, c\)) is a sequence with three terms.
While both sets and sequences perform a gathering role, there are several differences.
- The elements of a set are required to be distinct, but terms in a sequence can be the same. Thus, \((a, b, a)\) is a valid sequence of length three, but \(\{a, b, a\}\) is a set with two elements, not three.
- The terms in a sequence have a specified order, but the elements of a set do not. For example, \((a, b, c)\)and \((a, c, b)\) are different sequences, but \(\{a, b, c\}\) and \(\{a, c, b\}\) are the same set.
- Texts differ on notation for the empty sequence; we use \(\lambda\) for the empty sequence.
The product operation is one link between sets and sequences. A Cartesian product of sets, \(S_1 \times S_2 \times \cdots \times S_n\), is a new set consisting of all sequences where the first component is drawn from \(S_1\), the second from \(S_2\), and so forth. Length two sequences are called pairs.3 For example, \(\mathbb{N} \times \{a, b\}\) is the set of all pairs whose first element is a nonnegative integer and whose second element is an \(a\) or a \(b\):
\[\nonumber \mathbb{N} \times \{a, b\} = \{(0, a), (0, b), (1, a), (1, b), (2, a), (2, b), \ldots\}\]
A product of \(n\) copies of a set \(S\) is denoted \(S^n\). For example, \(\{0, 1\}^3\) is the set of all 3-bit sequences:
\[\nonumber \{0, 1\}^3 = \{(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)\}\]
3Some texts call them ordered pairs.