Sets (Computer Science Notes)
Sets
•
Definition: A set is an unordered collection of objects {x
∈
Z | 3 ≤ x ≤ 7}, first part tells
you that a variable x that ranges in the set and the second part gives you the constraints
for the variable, separator (| or :) is often read “such that.”
Things to be Careful About
•
A set is an unordered collection. So {1, 2, 3} and {2, 3, 1} are two names for the same
set. Each element occurs only once in a set. Or, alternatively, it doesn’t matter how many
times you write it. So {1, 2, 3} and {1, 2, 3, 2} also name the same set
•
We’ve seen ordered pairs and triples of numbers, such as (3, 4) and (4, 5, 2). The general
term for an ordered sequence of k numbers is a k-tuple.1 Tuples are very different from
sets, in that the order of values in a tuple matters and duplicate elements don’t magically
collapse (1, 2, 2, 3) 6= (2, 2, 1, 3) Pair() Set{}
•
∅ -
empty set
Cardinality, Inclusion
•
If A is a finite set (a set containing only a finite number of objects), then |A| is the number
of (different) objects in A. This is also called the cardinality of A
•
|{a, b, 3}| = 3 and |{a, b, a, 3}| is also 3, because we count a group of identical objects
only once
•
If A and B are sets, then A is a subset of B (written A
⊆
B) if every element of A is also in
B
•
The notion of subset allows the two sets to be equal. So A
⊆
A is true for any set A. So
⊆
is like ≤
•
If you want to force the two sets to be different (i.e. like <), you must say that A is a
proper subset of B, written A
⊂
B. You’ll occasionally see reversed versions of these
symbols to indicate the opposite relation B