2102 - Sets and Sequences

In this class, we will deal with some atomic values: true, false, and numbers. We will also deal with some composite values: sets and sequences. We call them composite because they are made up of other values.

Both sets and sequences

May contain any other value, including other sets and sequences
Are values, not entities
Are used to define other structures of interest in this course
Are commonly written with their contained values separated by commas

However, sets and sequences differ in several important ways:

Set	Sequence
Written with curly braces, like \{1,2\}	Written with parentheses, like (1,2)
Cannot contain the same value more than once; writing \color{darkred} \{1,1\} doesn’t make sense	Can contain the same value any number of times; (1,1) is a sequence and is distinct from (1) and (1,1,1)
Members have no order; \{2,3\} = \{3,2\}	Items have order; (2,3) \ne (3,2)
The number of members in a set is called its cardinality	The number of items in a sequence is called its length
The empty set (the only set with cardinality 0) is written \{\} or \emptyset	The empty sequence (the only sequence with length 0) is written () or \epsilon \varepsilon
Has many operators and special notations like \{1,2\} \cup \{x^2 \;\|\; x \in \mathbb N^{+}\}	Has no operators that are commonly used in computing
A singleton set is always distinct from its member; \{2\} \ne 2	A singleton sequence is often considered equal to its item; (2) = 2
Always called set	Called sequence or tuple, with special words for some lengths (e.g. pair, triple) and some element types (e.g. string)
Contained values are called members; element is also sometimes used.	Contained values are called items; elements is also sometimes used.