Course Content

Lecture – 1

00:36:16

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

01:06:28

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)