Relations and Functions

Understanding how one quantity relates to another is a key aspect of mathematics. We explore this through the important concepts of relations and functions. These ideas help us define, connect, and work with mathematical objects in a systematic way.

What is a set?

Before studying relations and functions, you must understand sets.

Cartesian products of sets

If A and B are two non-empty sets, the set of all ordered pairs (a,b), where \(a\in A \) and \(b\in B \) is called Cartesian product of A and B. Symbolically, we write
\((A\times B)={(a,b) : a\in A \ and \ b\in B}\).
Example: Let A={1,2} and B={x,y,z}, then
\(A\times B\)={(1,x),(1,y),(1,z),(2,x),(2,y),(2,z)}.

Diagrammatic representation of cartesian product of two sets

In order to represent \(A\times B \), by an arrow diagram, we first draw Venn diagrams representing sets A and B, one opposite to the other, as shown in the given figure, and write the elements of the sets. Now, we draw line segments starting from each element of set A and terminating to each element of set B. Example: Let A={1,2} and B={x,y,z}, then \(A\times B\)={(1,x),(1,y),(1,z),(2,x),(2,y),(2,z)}.
The following figure gives the arrow diagram of \(A\times B\).

Properties of cartesian products of sets

1. If two ordered pairs are equal, then the corresponding first elements are equal and the second elements are also equal. i.e., (a, b)=(x, y) then a=x and b=y
2. If n(A)=p and n(B)=q, then n(A\(\times \) B)=p\(\times \) q
3. \(A\times \phi =\phi \) and \(\phi \times A=\phi \)
4. \(A \times (B\cup C)= (A\times B) \cup (A\times C) \).
5. \(A \times (B\cap C)= (A\times B) \cap (A\times C) \)
6. \(A \times (B- C)= (A\times B) – (A\times C) \)
7. \(A\times B=B\times A \iff A=B \)

What is a Relation?

A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product \(A\times B \). The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in \(A\times B \). Basically, a relation is a connection between elements of two sets. Symbolically denoted as
\(R\subseteq A\times B \). Example: Let A={1,2,3} and B={a,b}, then \(A\times B \)={(1,a),(1,b),(2,a),(2,b),(3,a),(3,b)}.
Relation R could be {(1,a),(2,a)} etc.

Note: If the set A has m elements and the set B has n elements then the number of elements in the cartesian product of sets is mn elements and total number of relations is \(2^{nm}\).

Domain, Co-domain and Range

Domain: The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R. i.e., Domain of \(R=\{a:(a,b)\in f \}\).

Range: The set of all second elements of the ordered pairs in a relation R from a set A to a set B is called the range of the relation R. i.e., Range of \(R=\{b:(a,b)\in f \} \).

Co-domain: The whole set B of the relation R from a set A to a set B is called the co-domain. Note that range \(\subset \) co-domain.

Types of relations

1. Empty or void relation: As \(R=\phi \subset A\times A\) for any set A, so \(\phi \) is a relation on A, called the empty or void relation.
2. Universal relation: As \(R=A\times A \subseteq A\times A \), so R is a relation on A, called the universal relation.
3. Identity relation: The relation I_A ={(a,a): a\(\in \)A} is called the identity relation on A.

What is a function?

A function is a special type of relation where each element of the domain is associated with exactly one element of the co-domain.

A relation f from a set A to a set B is said to be a function if each element has a unique image in set B. In notation, \(f:A\rightarrow B\) means that f is a function from A to B, A is called the domain and B is called the co-domain of f.
If\( (a,b)\in f \Rightarrow f(a)=b \), Here, b is the image of a under f, and a is called the pre-image of b under f

Some specific types of functions

(i). Identity functions: Let R be the set of real numbers. A real-valued function f is defined as \(f:R\rightarrow R\) by y=f(x)=x for each value of \(x\in R \). Such a function is called the identity function.

(ii). Constant function: The function \(f:R\rightarrow R\) defined by f(x)=C for each \(x\in R\) is called a constant function. (Where C is a constant)

(iii.) Polynomial function: The function \(f:R\rightarrow R\) defined by \( y=f(x)=a_{0} +a_{1}x+a_{2} x^{2}+…+a_{n} x^{n} \) for each \(x\in R\) . where n is non- negative integer and \(a_{0}, a_{1}, a_{2},…,a_{n} \in R.\). Example: \(f(x)=x^{3}-3x \)

Some Important Questions on Relations and Functions

Question 1: If ( A = {1, 2} ) and ( B = {3, 4} ), find ( A \times B ) and the number of elements in it.
Solution:
\(A \times B = {(1,3), (1,4), (2,3), (2,4)}\)
\(n(A \times B) = n(A) \times n(B) = 2 \times 2 = 4\)

Question 2: Is the relation ( R = {(1, 2), (2, 3), (1, 3)} ) a function?
Solution: In a function, each element of the domain must have exactly one image of the co-domain. Here, the image of element 1 has 2 and 3, which violates the definition.\(\Rightarrow\) Not a function.

Question 3: How many functions can be defined from a set A with 3 elements to a set B with 4 elements?
Solution:
Total number of functions = \(n(B)^{n(A)}=4^3 \)= 64
Question 4: Let \( f: R \rightarrow R \) be defined by f(x) = 5 . Find domain, co-domain, and range.
Solution: Domain= R
Co-domain= R
Range=5
Question 5: If \( f(x) = x^2 + 1 \), find the range of f for \( x \in R \).
Solution: \(x^2 \geq 0 \Rightarrow x^2 + 1 \geq 1\)
Range\(= [1, \infty)\)
Question 6: Which of the following is a function?
(a) ( {(1,2), (2,3), (3,4)} )
(b) ( {(1,2), (1,3)} )
Solution:
(a) ✅ Function – each element of domain has unique image
(b) ❌ Not a function – 1 maps to two images.

Frequently asked questions on Relations and Functions

Q1. What is a relation in mathematics?
Answer: A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product \(A\times B \).

Q2. What is a function in mathematics?
Answer: A relation f from a set A to a set B is said to be a function if each element has a unique image in set B.

Q3. What is the Cartesian product of two sets?
Answer: If A and B are two non-empty sets, the set of all ordered pairs (a,b), where \(a\in A \) and \(b\in B \) is called the Cartesian product of A and B.

Q4. What is a constant function?
Answer: The function \(f:R\rightarrow R\) defined by f(x)=C for each \(x\in R\) is called a constant function. (Where C is a constant)