Relations and Functions (Part – 1)

This course is designed to help you gain a thorough understanding of fundamental mathematical concepts. Throughout your studies, you'll explore a wide range of topics, including Empty Relations, Universal Relations, Trivial Relations, Reflexive Relations, Symmetric Relations, Transitive Relations, Equivalence Relations, and Equivalence Classes.
To deepen your understanding of these concepts and prepare yourself for related examinations, this course includes a series of practice assignments. These assignments feature carefully selected questions from a variety of authoritative sources, including NCERT Textbook exercises, NCERT Examples, Board's Question Bank, RD Sharma, NCERT Exemplar, and more.
By engaging with these practice assignments, you'll develop a solid foundation in these essential mathematical concepts and be well-prepared to tackle any related challenges that come your way.

6 hours 11 minutes Course Duration
English Syllabus medium
Hindi + English Explanation
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Course Content

Lecture – 1


Topics Discussed:

  • {\displaystyle \text{Relation}\,\subseteq \,\,\text{Cartesian}\,\text{Product} }
  • Relation on set A is defined as {\displaystyle R\subseteq A\times A } .
  • {\displaystyle \phi } and {\displaystyle A\times A } are two extreme subsets of {\displaystyle A\times A } .
  • If {\displaystyle R=\phi } , then relation is called empty relation.
  • If {\displaystyle R=A\times A } , then relation is called universal relation.
  • Both empty and universal relations are also known as trivial relations.

Questions Discussed:

Question 1. Let R be the relation in the set N given by
R = {(a, b) : a = b ─ 2, b > 6}. Choose the correct answer.
A. (2, 4) ∈ R
B. (3, 8) ∈ R
C. (6, 8) ∈ R
D. (8, 7) ∈ R (N)

Question 2. Let A= {1, 2, 3} and define R = {(a, b): a ─ b = 12}. Show that R is empty relation on Set A. (B)

Question 3. Let A be the set of all students at a boy's school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R' = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation. (N)

Question 4. If A is the set of students at a school then write, which of following relations are Universal, Empty or neither of the two.
R1 = {(a, b) : a, b are ages of students and |a ─ b| > 0}
R2 = {(a, b) : a, b are weights of students, and |a ─ b| < 0}
R3 = {(a, b) : a, b are students studying in same class} (B)

Question 5. Let A = {1, 2, 3} and define R = {(a, b): a + b > 0}. Show that R is a universal relation on set A. (B)

Lecture – 5


Questions Discussed:

Question 6. Show that the relation R in the set A of points in a plane given by R = {(P, Q) : distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the circle passing through P with origin as centre. (N)

Question 7. Show that the relation R defined in the set A of all triangles as R=\{({{T}_{1}},\,{{T}_{2}}):\,{{T}_{{1\,}}}\,\text{is}\,\text{similar}\,\text{to}\,{{T}_{2}}\} is equivalence relation. Consider three right angle triangles {{T}_{1}} with sides 3, 4, 5, {{T}_{2}} with sides 5, 12, 13 and {{T}_{3}} with sides 6, 8, 10. Which triangles among {{T}_{1}},\,\,{{T}_{2}}\,\,\text{and}\,\,{{T}_{3}} are related? (N)

Question 8. Show that the relation R defined in the set A of all polygons as R = {(P1, P2): P1 and P2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5? (N)

Question 9. Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2) : L1 is parallel to L2 }. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4. (N)

Question 10. Let A = {0, 1, 2, 3, 4} and define a relation R on A as follows: R = {(0, 0), (0, 4), (1, 1), (1, 3), (2, 2), (3, 1), (3, 3), (4, 0), (4, 4)}. As R is an equivalence relation on A. Find the distinct equivalence classes of R. (P)

Question 11. Let R be the relation on set A=\{x\in Z:0\le x\le 10\} given by R = {(a, b) : (a-b ) is divisible by 4}. Show that R is an equivalence relation. Also, write all elements related to 4. (B)