ashish kumar

Ch1. Relations and Functions (Part - 1)

In this self study course, you will learn Empty Relations, Universal Relations, Trivial Relations, Reflexive Relations, Symmetric Relations, Transitive Relations, Equivalence Relations, Equivalence Classes. For further understanding of concepts and for examination preparation, practice questions based on the above topics are discussed in the form of assignments that have questions from NCERT Textbook exercise, NCERT Examples, Board’s Question Bank, RD Sharma, NCERT Exemplar etc. instead of only one book. The PDF of assignments can be downloaded within the course.

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6 hr 11 min 22 sec

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PDF of Notes, Assignments and Practice Test

Universal, Empty, Trivial Relations

• $\phi$ and $A\times A$ are two extreme subsets of $A\times A$ .
• If $R=\phi$ , then relation is called empty relation.
• If $R=A\times A$ , then relation is called universal relation.
• Both empty and universal relations are also known as trivial relations.

05:57

Assignment – 1

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Question – 1

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

02:50

Question – 2

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

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Question – 3

3. Let A be the set of all students of 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.

09:14

Question – 4

4. If A is the set of students of 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}

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Question – 5

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

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Reflexive, Symmetric, Transitive Relations

reflexive if $(a,\,\,a)\in \text{R},\,\forall \,a\in \text{A}$ or $a\text{R}a,\,\forall \,a\in \text{A}$
symmetric if $(a,\,\,b)\in \text{R}\,\Rightarrow \,(b,\,\,a)\in \text{R,}\,\,\forall \,\,a,\,b\,\in \text{A}$ or $a\text{R}b\,\Rightarrow \,b\text{R}a\text{,}\,\,\forall \,\,a,\,b\,\in \text{A}$
transitiveif $(a,\,\,b)\in \text{R,}\,\,(b,\,\,c)\in \text{R}\,\Rightarrow \,(a,\,\,c)\in \text{R, }\forall \,\,a,\,b,\,\,c\,\in \text{A}$

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Equivalence Relations and Equivalence Classes

• If a relation is reflexive, symmetric and transitive then the relation is said to be equivalence relation.
• An important property of an equivalence relation is that it divides the set into pairwise disjoint (or mutually disjoint) subsets called equivalence classes whose collection is called a partition of the set.
• The union of all equivalence classes gives the whole set.

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Assignment – 1

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Question – 6

6. Let A = {0, 1, 2, 3} and define a relation R on A as follows:
R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}.
Is R reflexive? symmetric? transitive?

09:09

Question – 7

7. Let R be the relation in the set {1, 2, 3, 4} given by
R = {(1, 2), (2, 2), (1, 1), (4,4), (1, 3), (3, 3), (3, 2)}.
Choose the correct answer. (N)
A. R is reflexive and symmetric but not transitive.
B. R is reflexive and transitive but not symmetric.
C. R is symmetric and transitive but not reflexive.
D. R is an equivalence relation.

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Question – 8

8. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

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Question – 9

9. Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

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Assignment – 2

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Question – 1 (9 Parts)

1. Determine whether each of the following relations are reflexive, symmetric and transitive:
A. Relation R in the set $\displaystyle A\text{ }=\text{ }\left\{ {1,\text{ }2,\text{ }3,\text{ }...,\text{ }13,\text{ }14} \right\}$ defined as
B. Relation R in the set N of natural numbers defined as $\displaystyle R=\left\{ {\left( {x,\text{ }y} \right):y=x+5\text{ and }x<4} \right\}$
C. Relation R in the set $\displaystyle A\text{ }=\text{ }\left\{ {1,\text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6} \right\}$ as R = {(x, y) : y is divisible by x}
D. Relation R in the set Z of all integers defined as R = {(x, y) : x – y is an integer}
E. Relation R in the set A of human beings in a town at a particular time given by
a. R = {(x, y) : x and y work at the same place}
b. R = {(x, y) : x and y live in the same locality}
c. R = {(x, y) : x is exactly 7 cm taller than y}
d. R = {(x, y) : x is wife of y}
e. R = {(x, y) : x is father of y}

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Assignment – 2

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Question – 2

2. Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y) : x and y have same number of pages} is an equivalence relation.

06:18

Question – 3

3. Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a – b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

16:09

Question – 4

4. Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.

02:23

Question – 5

5. Show that each of the relation R in the set $\displaystyle A=\{x\in Z:0\le x\le 12\}$ , given by
A. R = {(a, b) : |a – b| is a multiple of 4}
B. R = {(a, b) : a = b}
is an equivalence relation. Find the set of all elements related to 1 in each case.

22:11

Question – 6

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.

13:22

Question – 7

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?

06:42

Question – 10

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.

05:02

Question – 11

11. Let R be the relation on set $\displaystyle A=\{x\in Z:0\le x\le 10\}$ 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.

13:17

Question – 8

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?

12:13

Question – 9

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.

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Assignment – 2

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Question – 12

12. Let A = {1, 2, 3, .... , 12} and R be a relation in $A\times A$ defined by (p, q) R (r, s) if ps = qr $\forall$ $(p,\,\,q),\,\,(r,\,\,s)\,\in \,A\times A$ . Prove that R is an equivalence relation. Also obtain the equivalence class [(3, 4)].

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Question – 13

13. Let A = {1, 2, 3, ... 9} and R be the relation in A ×A defined by (a, b) R (c, d) if a + d = b + c for (a, b), (c, d) in A ×A. Prove that R is an equivalence relation and also obtain the equivalent class [(2, 5)].

12:20

Question – 14

14. Let N denote the set of all natural numbers and R be the relation on N $\times$ N defined by (a, b)R(c, d) if $ad(b+c)=bc(a+d)$ . Show that R is an equivalence relation.

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Before we continue

Something to discuss before we continue to the last three questions of the assignments.

04:55

Question – 15

15. Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.

24:45

Question – 16

16. Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a ≤ b2} is neither reflexive nor symmetric nor transitive.

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Question – 17

17. Check whether the relation R in R defined by R = {(a, b) : a ≤ b3} is reflexive, symmetric or transitive.

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