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A set is an unordered collection of items with no duplicates.
They are written in curly brackets, e.g. S = {123,
-5, 10.1}. Note that
{0, 0, 7}
is not a set because it contains two zeroes.
As we learnt in 11.01 (Number Systems), the symbol ℕ represents the
set of natural numbers, {0, 1,
2, 3, ...}, the symbol ℤ
represents the set of integers, {0, 1,
-1, 2, -2, ...},
and the symbols ℚ and ℝ represent the rationals and reals respectively.
The empty set, {}, can also be written using the symbol ∅.
The notation x ∈ S is used to check if an item x is in the set S.
We can use set comprehensions to compactly represent sets. Set comprehensions are of the form
{expression | condition}. For example, {3x | x ∈ ℕ} represents the
set {0, 3, 6,
9, ...}. In the condition, we can use the symbol ∧ to mean AND,
and the symbol ∨ to mean OR. For example, the comprehension {3x | x ∈ ℕ ∧ x ≥ 10}
represents the set {30, 33, 36,
39, ...}.
Sets can also contain strings. One way to write sets of strings is using notation such as
{AnBn | n ≥ 2}, which would mean "at least two As
followed by the same number of Bs", i.e. the set {AABB,
AAABBB, AAAABBBB, ...}.
A finite set is one which ends at some point.
This means that you can count its elements using natural numbers, up to a point. The length of
a finite set is called its cardinality. The set {123,
-5, 10.1} is finite with cardinality 3.
An infinite set is one which never ends.
For example, the natural numbers, integers, rational numbers, and real numbers.
A countably infinite set is one where every element of the set can be mapped to a different
natural number. The natural numbers are obviously countably infinite, as are the integers: an
example mapping could be {0: 0,
1: 1, 2:
-1, 3: 2,
4: -2, ...}. The rational numbers and real
numbers are uncountably infinite.
Write out the set represented by the comprehension {PmQn | m + n = 7}.
{PPPPPPP, PPPPPPQ, PPPPPQQ,
PPPPQQQ, PPPQQQQ, PPQQQQQ,
PQQQQQQ, QQQQQQQ}