NIMCET Relations and Functions: Formulas & Practice Questions

NIMCET Relation and Functions is an important topic in the Mathematics section of the exam. Questions from this chapter can be based on Cartesian products, relations, domain, codomain, range and different types of functions. Some questions may also ask you to find the number of possible relations or functions using direct formulas.

Before solving questions, make sure you understand how two sets are connected and how a function is different from a general relation. Once these basics are clear, you can solve most NIMCET Relation and Functions questions more confidently and in less time.

Cartesian Product of Sets

The Cartesian product of two sets is the set of all ordered pairs where the first element is from the first set and the second element is from the second set. If 

A

A and 

B

B are two sets, their Cartesian product is denoted as 

A×B

A×B and consists of all pairs 

(a,b)

(a,b) such that 

a∈A

a∈A and 

b∈B

b∈B.

• Definition: Relation

A relation shows how the elements of one set are connected to the elements of another set. It is formed using ordered pairs from the Cartesian product of two sets.

A function is a special type of relation in which every element of the first set (domain) is connected to exactly one element of the second set (codomain).

• A×B={(a,b)∣a∈A,b∈B}

• A×B={(a,b)∣a∈A,b∈B}

• Example: If 

• A={1,2,3,4}

• A={1,2,3,4} and 

• B={5,6,7}

• B={5,6,7}, then 

• A×B

• A×B contains all pairs like 

• (1,5),(1,6),(1,7),(2,5),…,(4,7)

• (1,5),(1,6),(1,7),(2,5),…,(4,7).

• Number of Elements: If 

• ∣A∣=m

• ∣A∣=m and 

• ∣B∣=n

• ∣B∣=n, then 

• ∣A×B∣=m×n

• ∣A×B∣=m×n.

Definition and Nature of Relations

A relation from set 

A

A to set 

B

B is any subset of the Cartesian product 

A×B

A×B. It can be the entire 

A×B

A×B, a proper subset, or even the empty set.

• Definition: A relation 

• R

• R from 

• A

• A to 

• B

• B is a subset of 

• A×B

• A×B.

• Notation: 

• a

• a is related to 

• b

• b (written as 

• aRb

• aRb) if 

• (a,b)∈R

• (a,b)∈R.

• Example: If 

• A={1,2,3}

• A={1,2,3} and 

• B={3,4,5,6}

• B={3,4,5,6}, then 

• A×B

• A×B includes pairs like 

• (1,3),(1,4),…,(3,6)

• (1,3),(1,4),…,(3,6). A relation could be all pairs where 

• a<b

• a<b.

Examples of Relations

Relations can be defined by specific conditions on the elements of 

A

A and 

B

B. Several examples illustrate how relations are subsets of 

A×B

A×B based on such conditions.

• Relation 1: 

• a>b

• a>b (all pairs where the first element is greater than the second)

• Relation 2: 

• a<b

• a<b (all pairs where the first element is less than the second)

• Relation 3: 

• a≤b

• a≤b (all pairs where the first element is less than or equal to the second)

• Relation 4: Second element is twice the first (

• b=2a

• b=2a)

• Relation 5: Difference is 2 (

• a−b=2

• a−b=2)

• Null Relation: A relation with no elements (e.g., if no pair satisfies the given condition)

Special Cases and Self-Relations

A relation can also be defined on a set with itself, i.e., 

A×A

A×A. This is called a self-relation.

• Example: If 

• A={1,2,3,…,32}

• A={1,2,3,…,32}, a relation could be all pairs 

• (a,b)

• (a,b) where 

• a

• a is one-fourth of 

• b

• b.

• Construction: For 

• a

• a in 

• A

• A, include 

• (a,4a)

• (a,4a) if 

• 4a

• 4a is also in 

• A

• A.

Counting Relations

The total number of possible relations from 

A

A to 

B

B depends on the number of elements in 

A×B

A×B. Since each subset of 

A×B

A×B is a relation, the count is based on the power set.

• Formula: If 

• ∣A∣=m

• ∣A∣=m and 

• ∣B∣=n

• ∣B∣=n, then the number of relations from 

• A

• A to 

• B

• B is 

• 2mn

• 2

• mn

• .

• For Self-Relations: If 

• ∣A∣=m

• ∣A∣=m, then the number of relations on 

• A

• A is 

• 2m2

• 2

• m

• 2

• .

Example: If 

• ∣A∣=3

• ∣A∣=3 and 

• ∣B∣=4

• ∣B∣=4, then the number of relations is 

• 212

• 2

• 12

Domain, Codomain, and Range of a Relation

The domain, codomain, and range are key concepts associated with a relation.

Domain: The set of all first elements (from 

• A

• A) that appear in the ordered pairs of the relation.

Codomain: The entire set 

• B

• B (second set in 

• A×B

• A×B).

Range: The set of all second elements (from 

• B

• B) that actually appear in the relation.

Example: If 

• R={(4,1),(5,2),(6,3)}

• R={(4,1),(5,2),(6,3)}, then:

• Domain of 

• R

• R is 

• {4,5,6}

• {4,5,6}

• Range of 

• R

• R is 

• {1,2,3}

• {1,2,3}

• Codomain is the full set 

• B

• B from which the second elements are taken

Worked Examples and Application Questions

Several examples illustrate how to construct relations and determine their properties.

Example 1: Relation defined on natural numbers where 

• a−b=3

• a−b=3. Only pairs like 

• (4,1),(5,2),(6,3),…

• (4,1),(5,2),(6,3),… are included, as subtraction must yield 3 and both 

• a,b

• a,b are natural numbers.

Example 2: Relation where the second element is the cube of the first plus one, i.e., 

• y=x3+1

• y=x

• 3

• +1. For 

• x

• x in 

• A

• A, compute 

• y

• y and include 

• (x,y)

• (x,y) if 

• y

• y is in 

• B

• B.

• Domain and Range Extraction: For a relation 

• R={(1,2),(3,28),(4,65),(5,126),(6,227)}

• R={(1,2),(3,28),(4,65),(5,126),(6,227)}, domain is 

• {1,3,4,5,6}

• {1,3,4,5,6}, range is 

• {2,28,65,126,227}

• {2,28,65,126,227}, and codomain is the full set 

• B

• B.

Key Results and Formulas

Several important results and formulas are directly stated and used for problem-solving.

• Number of Elements in Cartesian Product: 

• ∣A×B∣=∣A∣×∣B∣

• ∣A×B∣=∣A∣×∣B∣

• Number of Relations: 

• 2∣A×B∣=2mn

• 2

• ∣A×B∣

• =2

• mn

• , where 

• ∣A∣=m

• ∣A∣=m, 

• ∣B∣=n

• ∣B∣=n

• For Self-Relations: 

• 2m2

• 2

• m

• 2

• , where 

• m=∣A∣

• m=∣A∣

• Power Set Cardinality: The number of subsets of a set 

• A

• A is 

• 2∣A∣

• 2

• ∣A∣

Practical Interpretation of Relations

Relations can be visualized and interpreted in real-life contexts, such as mapping players to teams or actors to movies.

• Example: Mapping "Virat Kohli" to "RCB", "MS Dhoni" to "CSK", and "Rohit Sharma" to "MI" forms a relation between the set of players and the set of teams.

• Another Example: Mapping actors to movies, e.g., "Ranbir Kapoor" to "Animal", "Sunny Deol" to "Gadar 2", "Shah Rukh Khan" to "Jawan".

Exam-Focused Problem Types

Typical exam questions involve constructing relations, counting the number of possible relations, and identifying domain, range, and codomain.

• Constructing Relations: Given a condition (e.g., 

• a<b

• a<b, 

• b=2a

• b=2a), list all pairs satisfying it.

• Counting Relations: Use the formula 

• 2mn

• 2

• mn

•  for total relations.

• Domain/Range Identification: Extract first and second elements from the relation's pairs.

• Special Cases: Null relation (no pairs), full relation (

• A×B

• A×B), and relations defined by arithmetic conditions.

Important Observations and Pitfalls

Understanding the difference between domain, range, and codomain is crucial. The range may not cover the entire codomain, and the domain may not cover all of 

A

A.

• Null Relation: If no pair satisfies the condition, the relation is empty.

• Full Relation: If all pairs are included, the relation equals 

• A×B

• A×B.

• Counting Error: Always use the correct formula for the number of relations; do not confuse with the number of functions or mappings unless specified.

• For any relation, always check the specific condition given and construct the subset accordingly.

• When extracting domain and range, list only those elements that actually appear in the relation, not the entire set.

• For exam questions, focus on the explicit condition defining the relation and apply the formulas as stated above.

Benefits of Studying Relation and Functions for NIMCET

Preparing for Relations and Functions can help you handle different types of Mathematics questions with more confidence. The chapter also has several direct concepts and formulas that can save time during the exam.

Some key benefits of preparing this topic are:

• Helps with formula-based questions: Many questions can be solved quickly once you remember the formulas for Cartesian products, relations and functions.

• Improves conceptual clarity: Understanding domain, codomain, range and different types of relations makes it easier to identify what a question is asking.

• Saves time during the exam: Questions based on counting relations or identifying domain and range can often be solved in a few steps.

• Useful for practice: The chapter includes different question types, so regular practice can improve your speed and accuracy.

• Builds a base for functions: A clear understanding of relations makes it easier to study functions and their different types.

How to Solve NIMCET Relation and Functions Questions

The approach depends on the type of question. First, identify the concept being tested and then apply the relevant definition or formula.

Step 1: Identify the given sets

Read the question carefully and note the sets A and B and the number of elements in each set. If the question gives |A| = m and |B| = n, keep these values ready for applying the formula.

Step 2: Check whether it is a Cartesian Product or Relation

If the question asks for all possible ordered pairs, use the Cartesian product:

A × B = {(a, b) | a ∈ A, b ∈ B}

If the question gives a condition such as a < b, a > b or b = 2a, select only the ordered pairs that satisfy that condition.

Step 3: Use the correct formula

For a question asking for the total number of relations from A to B, use:

Number of relations = 2^(mn)

where m = |A| and n = |B|.

For example, if A has 2 elements and B has 3 elements:

Number of relations = 2^(2 × 3) = 2⁶ = 64

Do not confuse this formula with the formula for the number of functions.

Step 4: Find the domain, codomain and range

For a relation written as ordered pairs:

Domain: Take the first elements.

Range: Take the second elements that actually appear.

Codomain: Take the complete second set given in the question.

For example, if

R = {(1, 3), (2, 4), (3, 4)}

then:

• Domain = {1, 2, 3}

• Range = {3, 4}

• Codomain depends on the complete set B given in the question.

Remember that the range can be smaller than the codomain.

Step 5: Check the type of relation or function

If the question asks you to identify a type of relation or function, check the given condition carefully. For functions, make sure every element of the domain has exactly one image in the codomain.

Step 6: Practise different question types

After learning the formulas, solve questions based on Cartesian products, counting relations, domain and range, types of relations and functions. This will help you recognise the pattern of a question quickly during the NIMCET exam.

Relations and Functions becomes easier when the basic concepts and formulas are clear. For NIMCET, focus on Cartesian products, types of relations, domain, codomain, range and the difference between relations and functions. Revise the important formulas regularly and practise different types of questions to improve your calculation speed and accuracy.