Relations and Functions
Definition 1 (Cartesian Product) The Cartesian product of two sets S1 and S2 is the
set
S1 × S2 = (a, b) a ∈ S1 , b ∈ S2
The set of ordered pairs in a set S is S × S.
Example 2 If S1 = {1, 2, 3} and S2 = {4, 5}, then the Cartesian product
S1 × S2 =
(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)
and the set of ordered pairs in S2 is
S2 × S2 =
(4, 4), (4, 5), (5, 4), (5, 5)
Definition 3 (Relation) A relation in a set S is a set R of ordered pairs in S. That is,
a relation is a subset R ⊂ S × S. Often one writes aRb instead of (a, b) ∈ R.
Example 4 If S = {1, 2, 3}, then the normal “strictly smaller than” relation is
{ (1, 2), (1, 3), (2, 3) }, which is usually written 1 < 2, 1 < 3, 2 < 3.
Definition 5 (Order Relation) An order relation on a set S is a relation, denoted by
<, on S that obeys
(i) If x ∈ S and y ∈ S then exactly one of the statements
x < y,
x = y,
y<x
is true.
(ii) If x, y, z ∈ S and x < y and y < z, then x < z. This is called transitivity.
One often writes y > x in place of x < y and x ≤ y in place of “x < y or x = y”. An
ordered set is a set together with an order relation.
Example 6 The set of rational numbers Q is an ordered set with the order relation
a < b ⇐⇒ b − a =
c Joel Feldman.
2014. All rights reserved.
m
n
for some m, n ∈ IN
August 7, 2014
Relations and Functions
Definition 7 (Equivalence Relation) A relation R on a set S is said to be
(i) reflexive if a ∈ S implies (a, a) ∈ R
(ii) symmetric if (a, b) ∈ R implies (b, a) ∈ R
(i) transitive if (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R
An equivalence relation on a set S is a relation which is reflexive, symmetric and transitive.
One often writes a ∼ b in place of (a, b) ∈ R.
If ∼ is an equivalence relation on a set S, then the equivalence class of a ∈ S (under ∼) is
the set of all elements of S that are equivalent to a. It is denoted [a]. That is
[a] = s ∈ S s ∼ a
The set of all equivalence classes in S is called the quotient set of S by ∼. It is denoted
S/ ∼= [a] a ∈ S
Example 8 Suppose that you have just defined the integers ZZ, and you now want to
define the rational numbers. A natural first guess would be the set of all pairs (a, b) of
integers with b 6= 0. That is, the Cartesian product Q = ZZ × (ZZ \ {0}). Here a is to play
the role of the numerator and b is to play the role of the denominator. This does not quite
work because, for example (2, 1) and (4, 2) are two different elements of Q, but we are
thinking of them both as being 2 = 12 = 42 . So we define an equivalence relation on Q by
(a, b) ∼ (c, d) ⇐⇒ ad = bc
(Go ahead and check that this is really an equivalence relation.) We would then think of
a rational number as an equivalence class [(a, b)] and define the set of all rational numbers
as the quotient set Q = Q/ ∼. Addition and multiplication would be defined by
[(a, b)] + [(c, d)] = [(ad + bc, bd)]
[(a, b)] [(c, d)] = [(ac, bd)]
Of course there are many (simple) things to be checked, to verify that these definitions
make sense and obey the usual arithmetic axioms. But this really is the standard way to
define the rational numbers.
Definition 9 (Function) A function F with domain D and range R, denoted F : D → R,
is a subset F ⊂ D × R with the properties, first, that for each x ∈ D there is some y ∈ R
with (x, y) ∈ F and, second, that
(x, y) ∈ F, (x, z) ∈ F =⇒ y = z
That is, F assigns to each element of the domain D a unique element of the range R.
Usually one writes y = F (x) in place of (x, y) ∈ F .
c Joel Feldman.
2014. All rights reserved.
August 7, 2014
Relations and Functions