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Jenny C. Cano, LPT, MSc
MODULE 3
Relations and Functions
Relation
We are acquainted with different relationships here in the world. In the world of mathematics,
relationships do have its place also. As an example, in our previous discussions, we have the
relationships among sets, sequences, equality and logical equivalence. However, instead of using the
word relationships in describing these terms, we use the word relations. Generally, we can think of a
relation as a list of relationship of elements to other elements. It can be thought as a set of ordered
pairs. The formal definition of a relation follows.
Definition 3.1 : A (binary) relation R between set A and set B is a subset of the Cartesian product 𝐴 𝑥 𝐵.
If (𝑥, 𝑦) ∈ 𝑅, then we write 𝑥𝑅𝑦 and say that 𝑥 is related to 𝑦. If 𝐴 = 𝐵, we call R a (binary) relation on
A.
The set
is called the domain of R. The set
is called the range of R.
Examples:
- Let 𝐴 =
and 𝐵 =
. If we define a relation R from A to B by (𝑥, 𝑦) ∈ 𝑅 where 𝑥
divides 𝑦, we obtain
- Let R be the relation on 𝐴 =
defined by (𝑥, 𝑦) ∈ 𝑅 where 𝑥 − 𝑦 = 0 , 𝑥, 𝑦 ∈ 𝐴. Then,
- Let R be the relation on 𝐴 =
defined by (𝑥, 𝑦) ∈ 𝑅 where 𝑦 = 𝑥 − 1 , 𝑥, 𝑦 ∈ 𝐴. Then,
The domain of R is the set
and the range of R is the set
- What are the domain and range of R in #1?
An informative way to picture a relation on a set is to draw its digraph. (More detailed discussions
of digraphs will be given in the following module). We will only tackle now digraphs in connection with
relations.) Digraph is a shorthand term of directed-graph. To draw a digraph of a relation on a set A, we
first draw dots or vertices to represent the elements of A. If the element (𝑥, 𝑦) is in the relation, we
draw an arrow (called directed edge ) from 𝑥 to 𝑦. Note that an element of the form (𝑥, 𝑥) in a relation
corresponds to a directed edge from 𝑥 to 𝑥. Such an edge is called a loop.
Jenny C. Cano, LPT, MSc
Examples:
- The relation 𝑅 =
on 𝐴 =
can be described by the
following digraph.
- The relation 𝑅 =
on 𝐴 =
can be described by the
following digraph.
Note: The elements of the given set can be written in any order or position as long as the
relation is maintained. That is, the first element of the ordered pair should always point the
second element. One ordered pair corresponds to one arrow. So, the number of arrows in the
digraph is the number of ordered pairs in the relation.
Properties of Relations
Let R be a (binary) relation on set A. Then,
1. R is said to be reflexive if (𝑥, 𝑥) ∈ 𝑅 for every 𝑥 ∈ 𝐴. 2. R is said to be symmetric if for all 𝑥, 𝑦 ∈ 𝐴, if (𝑥, 𝑦) ∈ 𝑅, then (𝑦, 𝑥) ∈ 𝑅. 3. R is said to be antisymmetric if for all 𝑥, 𝑦 ∈ 𝐴, if (𝑥, 𝑦) ∈ 𝑅, then (𝑦, 𝑥) ∈ 𝑅. 4. R is said to be transitive if for all 𝑥, 𝑦, 𝑧 ∈ 𝐴, if (𝑥, 𝑦) ∈ 𝑅 and (𝑦, 𝑧) ∈ 𝑅, then (𝑥, 𝑧) ∈ 𝑅.
Examples:
- The relation 𝑅 =
on 𝐴 =
is not reflexive since 2 ∈ 𝐴
but ( 2 , 2 ) ∈ 𝑅. 𝑅 is not symmetric since ( 1 , 2 ) ∈ 𝑅 but ( 2 , 1 ) ∈ 𝑅. 𝑅 is not antisymmetric since
( 3 , 4 ) ∈ 𝑅 and ( 4 , 3 ) ∈ 𝑅. 𝑅 is not transitive since
, ( 4 , 3 ) ∈ 𝑅 but
- The relation 𝑅 =
on 𝐴 =
is not reflexive, not symmetric,
not transitive, but antisymmetric.
Jenny C. Cano, LPT, MSc
Examples:
}is a partition of set 𝐴 =
- The sets 𝐹 =
, and 𝐻 =
form a partition of set 𝐴 =
Note: If set 𝐴 has a partition 𝐶, then every element of 𝐴 should be contained to exactly one member of
Definition 3.5 : Let 𝐶 be a partition of a set 𝐴. Define 𝑥𝑅𝑦 to mean that for some set 𝐹 in 𝐶, both 𝑥 and
𝑦 belong to 𝐹. Then, 𝑅 is reflexive, symmetric, and transitive.
Note: A relation that is reflexive, symmetric, and transitive is called equivalence relation. That is, the
relation that will be derived on any set from its partition is an equivalence relation.
Examples:
- Consider the partition
of 𝐴 =
. The relation 𝑅 on 𝐴 given by definition 3.5 follows.
- Consider the partition
of 𝐴 =
. What is 𝑅?
Definition 3.6 : Let 𝑅 be a equivalence relation on set 𝐴. For each 𝑥 ∈ 𝐴, let
[𝑥] = {𝑦 ∈ 𝐴: 𝑦𝑅𝑥}
Then,
𝐶 = {[𝑥]: 𝑥 ∈ 𝐴}
is a partition of 𝐴. The sets
[
]
are called the equivalence classes of 𝐴 given by the relation 𝑅.
Examples:
- Let the relation
on 𝐴 =
be an equivalence relation. The equivalence class
[
]
containing 1 consists
of all 𝑥 such that (𝑥, 1 ) ∈ 𝑅. Therefore,
[
]
The remaining equivalence classes are found similarly:
[
]
[
]
[
]
[
]
[
]
- Consider the partition
on 𝐴 =
. Then, the following equivalence classes are found:
[ 0 ] = [ 2 ] = [ 3 ] = { 0 , 2 , 3 }, [ 1 ] = { 1 }, [ 4 ] = { 4 }.
- Given the equivalence relation
on the set
. What are the equivalence classes?
Jenny C. Cano, LPT, MSc
Function
Definition 3.7 : Let 𝐴 and 𝐵 be sets. A function 𝒇 from 𝐴 to 𝐵 is a subset of the Cartesian product 𝐴 𝑥 𝐵
having the property that for each 𝑥 ∈ 𝐴, there is exactly one 𝑦 ∈ 𝐵 with (𝑥, 𝑦) ∈ 𝑓.
𝑓 is a set of ordered pair. We sometimes denote a function 𝑓 from 𝐴 to 𝐵 as 𝑓: 𝐴 → 𝐵. The set 𝐴 is
called the domain of 𝑓 and the set
(which is a subset of 𝐵) is called the range of 𝑓. The set 𝐵 is sometimes called as the codomain of 𝑓.
Note that a function is a special kind of relation. All functions are relation but not all relations are
functions. From the above definition, it is very clear that for a function 𝑓: 𝐴 → 𝐵 all the elements of the
set 𝐴 must have exactly one pair in the set 𝐵. If (𝑥, 𝑦) ∈ 𝑓, then we call 𝑦 the image of 𝑥 under 𝑓 and
we write 𝑓
Examples: Given the sets 𝐴 =
and 𝐵 =
. Then,
1
is not a function since 1 has two different pairs.
2
is a function.
3
= {( 1 , 𝑎)} is not a function from A to B since 2 has no pair.
- How about 𝐴 𝑥 𝐵?
The situation of classifying a set to be a function or not can be depicted by drawing a schematic or
arrow diagram. For an arrow diagram to be a function there should only be exactly one arrow from
each element in the domain. Consider the following examples.
𝐴 𝐵 C D
Fig. 3.7.1 Fig. 3.7.
Fig. 3.7.1 is a function from A to B since all the elements in the domain (set 𝐴) has exactly one
arrow. Fig. 3.7.2 is not a function from C to D since 3 ∈ 𝐴 has no arrow. Another case wherein the
arrow diagram is not a function is when one or more elements in the domain have two or more
arrows. Consider the following cases:
- The set
is not a function from 𝐴 =
to 𝐵 =
because the element 4 in 𝐴 is not assigned
to an element in 𝐵.
1
2
3
a
b
c
1
2
3
4
a
b
c
Jenny C. Cano, LPT, MSc
The condition given in definition 3.8 is equivalent to the following statements:
1
2
∈ 𝐴 if 𝑓
1
2
), then 𝑥
1
2
1
2
∈ 𝐴 if 𝑥
1
2
, then 𝑓(𝑥
1
2
The negation of definition 3.8 is given by the following statement:
➢ 𝑓 is not injective if ∃𝑥
1
2
1
2
but 𝑓
1
2
Examples:
- The function
from 𝐴 = { 1 , 2 , 3 } to 𝐵 = {𝑎, 𝑏, 𝑐, 𝑑} is one-to-one.
- The function
is not one-to-one since 𝑓( 1 ) = 𝑎 = 𝑓( 3 ).
- The function
from the set of positive integers to the set of positive integers is one-to-one. The proof follows.
We will show that for all positive integers 𝑛
1
and 𝑛
2
, if 𝑓
1
2
), then 𝑛
1
2
Let 𝑓
1
2
). Then, 2 𝑛
1
2
Thus,
1
2
1
2
1
2
Therefore, 𝑓 is one-to-one.
- The function
𝑛
2
from the set of positive integers to the set of integers is not one-to-one. The proof follows.
We must find positive integers 𝑛
1
and 𝑛
2
1
2
, such that 𝑓
1
2
). Suppose 𝑛
1
and 𝑛
2
= 4. Then, 𝑓
2
2
= 0 and 𝑓
4
2
= 0. Thus, 𝑓
Therefore, 𝑓 is not one-to-one.
Definition 3.9 : If 𝑓 is a function from 𝐴 to 𝐵 and the range of 𝑓 is 𝐵, 𝑓 is said to be onto 𝐵 (or an onto
function or a surjective function). In symbol, 𝑓 is surjective if ∃𝑥 ∈ 𝐴 𝑠. 𝑡. 𝑓
= 𝑦, ∀𝑦 ∈ 𝐵. Also, 𝑓 is
surjective if 𝑓
The negation of definition 3.9 is given by the following statement:
➢ 𝑓 is not surjective if ∃𝑦 ∈ 𝐵 𝑠. 𝑡. ∀𝑥 ∈ 𝐴, 𝑓
Examples:
- The function
from 𝐴 =
to 𝐵 =
is one-to-one and onto 𝐵.
Jenny C. Cano, LPT, MSc
- The function
from 𝐴 = { 1 , 2 , 3 } to 𝐵 = {𝑎, 𝑏, 𝑐, 𝑑} is not onto 𝐵.
- The function pictured by an arrow diagram in Fig. 3.7.1 is neither one-to-one nor onto 𝐵.
- The function
from the set of natural numbers to the set of natural numbers is not surjective. The proof
follows. We will show that 𝑓
Thus, 𝑓(𝐴) ≠ 𝐵. Therefore, 𝑓 is not surjective.
- The function
2
from 𝐴 (set of real numbers) to 𝐵 (set of positive real numbers including zero) is onto 𝐵. The
proof follows.
2
Thus, 𝑓
= 𝐵. Therefore, 𝑓 is onto 𝐵.
Definition 3.10 : A function that is both injective and surjective is called a bijection.
Exercises:
- Show that the function
3
from the set of real numbers to the set of real numbers is a bijection.
- Using vertical line test, verify whether the following are functions or not:
a. 𝑓
2 − 3 𝑥
5
b. 𝑥 = 𝑦
2
Definition 3.11 : Let 𝑓: 𝐴 → 𝐵 be a bijection. The inverse of 𝑓, denoted by 𝑓
− 1
, is the set
− 1
Jenny C. Cano, LPT, MSc
- Let 𝑓
2
and 𝑔
= 5 𝑥 + 2. Then,
2
2
2
2
What is 𝑓(𝑔
Exercises: Do what is asked.
- Let 𝑓
𝑥
2
2
. Then, find 𝑓(− 1 ) and 𝑓 (
1
2
- Find the inverse of the following:
a. 2 𝑥 − 𝑦 = 1
b. 3 𝑥 = 2 𝑦 − 5
c. 0 = 𝑦 − 𝑥
Jenny C. Cano, LPT, MSc
Worksheet 4
Name: ____________________________________ Score: _________
Date: ________________
Instruction: For your answers, you can use the back portion of this paper or in the extra sheet.
I. Refer the following (numbers 1-6) to the relation 𝑅 on the set { 1 , 2 , 3 , 4 , 5 } defined by the
rule
∈ 𝑅 if 3 divides 𝑥 − 𝑦.
- List the elements of 𝑅.
- Find the domain of 𝑅.
- Find the range of 𝑅.
- List the elements of 𝑅
− 1
- Identify the properties of 𝑅.
- Draw the digraph of 𝑅
− 1
II. Determine whether the given relation is an equivalence relation on
. If the relation
is equivalence relation, list the equivalence classes.
2
III. Verify whether the function
is a bijection from the set of real numbers to the set of real numbers. If 𝑓 is a bijection, find
its inverse.
IV. Let 𝑓
2
− 4 𝑥 + 1 and 𝑔
𝑥
. Find the following:
1
2
