maths chapter 1.pdf. maths class 12th chapter 1

Maths CH 1 12th

1. RUZ PODAR INTERNATIONAL SCHOOL
fh) RAJKOT (CBSE)

CLASS : XII
CHAPTER 1

\ RELATIONS al FUNCTIONS )

Prepared by : Vaibhav Rupareli VR

2. INTRODUCTION:

A 8

All possible relations from set A to B
AXB=((14),(15), (4,6), (24), (25), (26), (34),(35), 66)

A

example, R: A» B defined as
(vy)ysxt3VxeAyeB)
(25), (3,6)} i

R=(
R=((,,05),

3. Relations

rsal Relation
(largest relation)

(males relation)

R:A>B
R=AxB

Eg: R={(x,y)sx>y¥x€Ay €B} Eg.R=((1,y):x<yVx€4yEB)

A B

4. a

Q-1 Let A be the set of all students of a boys school. Check the relation R in A
given by

(i) R= {(a, b) : a is sister of b}

(ii) R' = {(a, b) : the difference between heights of aand bis less than 3 meters} .

Solution

5. ag

Q-1 Let A be the set of all students of a boys school. Check the relation R in A
given by

(i) R= ((a, b) : ais sister of bj

(ii) R’ = {(a, b) : the difference between heights of a and bis less than 3 meters} .

Solution: For relation R

R= ((a, b) ¿a is sister of b}

Itis a boys school, so there are no girls student.

There cannot be sister of any student of the school.

Hence, R =, ¡.e. Ris an empty relation. Vr

6. 00%

R'= {(a, b) : the difference between heights of a and b is less

For relation R'

than 3 meters)

The difference between heights of any two students of the school

has to be less than 3 meters (9,8 foot).

So, R' is the universal relation,

7. Identity & Reflexive Relation

LetR: A> Abe a relation on set A = {1, 2,3}

yp A = YY

S
Reflexive relation

Identity relation

“When alla€ Ainrelatedto / { When (a,a) ERforalla € A A

i |
uam | \ example, R= ((11),(12),(22).@3)|

"Example, j= ((,0,020,,33))) / D Lite MAY
(AI) Doublets only] other pair can be there] +

8. LetR : 4 > Abe a relation on set 4 = (1,2,3)

When v (a,b) € R there exists (b,a) € R

Example, R = ((1,2),(2,3),(3,2),(2,1)

9. ‘Transitive Relation

LetR: A - Abearelation on set A = {1, 2,3}

If (a,b) € Rand (b,c) € R then (a,c) ER

Example, R = {(1,2), (2,3)(1,3)} Transitive 7

10. tion which is
For example,

symmetric

and transitive Ry = {(1,1), (2,2), (3,3), (1,2),(2,1)} on A = (1,2,3)
Ry = {(1,1), (2,2), 3,3), (2,3), (3,2)} on A = (12,3)

11. Equivalence class is the name given to a subset
of some equivalence relation R which includes
all the elements that are equivalent to each
other.

[x] = {yl(x, y)€R}

SN Equivalence class of xeA

12. 00%

Q-2 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.

Solution

13. 00%

Q-2 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.

Solution
Check Reflexive Check symmetric
If the relation is reflexive, To check whether symmetric or not,

then (a, a) ER for every a €f1,2,3} lf(a, b) ER, then (b, a) ER

Here (1,2) ER, but (2, 1) ER

Since (1,1) E R (2, 2) ER 8. (3, 3) ER

“Ris not symmetric
“Ris reflexive VR

14. 00%

To check whether transitive or not,
If (a,b) ER & (b,c) ER, then (a,c) ER

Check transitive

Here, (1, 2) € Rand (2,3) € Rbut (1, 3) ER.

+. Ris not transitive

Hence, R is reflexive but neither symmetric nor transitive.

15. 00%

Q-3 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.

solution

16. 00%

Q-3 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.

Solution 4=$1.23)
TUN

R=((1,2),(2,1))

(1,0,(2,2),(3,3) 2 R

“+ Ris not reflexive.

(1,2) e Rand (21) eR
¿R is symmetric.

(1,2) € Rand (2,1) e R

(LI) eR
. Ris not transitive.

R is symmetric, but not reflexive or transitive. VR

17. 00%

Q-4 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.

solution

18. Q-4 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.

Solution
4=(1,2,3,4,5,6) 0 :
“. R'isnot symmetric.
R=¿(a,b):b=a+1)
R=1(1,2),(2,3),(3,4),(4,5),(5,6)]_-(1,2),(2,3) e R
(1,3) e R
(a,a) e R,a € A “+ Ris not transitive.
(1,1),(2,2),(3,3),(4,4),(5,5) € R
: R isnot reflexive. R is neither reflective nor symmetric nor transitive.

(1,2)eR, but (2.1) R Vr

19. 00%

Q-5 Determine whether each of the following relations are reflexive, symmetric
and transitive:

Relation R in the set N of natural numbers defined as

R=((x y) : y =x+5 and x< 4}

Solution

20. Q-5 Determine whether each of the following relations are reflexive, symmetric
and transitive:

Relation R in the set N of natural numbers defined as

R=((x y) : y =x+5 and x< 4}

Solution
R-{(L6)(27) 6)
R is not reflexive because (1,1) ER .

R is not symmetric because (1,6) € R but (6,1 Jer.
R is not «transitive because there isn’t any ordered pair in R such that
(x,y),(2)eR, so (nz)eR,

Hence, R is neither reflexive nor symmetric nor transitive.

21. 00%

Q-6 Determine whether each of the following relations are reflexive, symmetric
and transitive:

Relation R in the set A = (1, 2, 3, ..., 13, 14) defined as

R = {(x, y) : 3x— y = 0)

Solution

22. Q-6 Determine whether each of the following relations are reflexive, symmetric
and transitive:

Relation R in the set A = {1, 2, 3, ..., 13, 14} defined as
R= {(x, y) : 3x-— y = 0}

Solution
R={(1,3),(2,6),(3,9),(4,12)}
R is not reflexive Because (L!),(2,2)... and (14,14) ¢R.

R is not symmetric because (1,3)E R, but (3,1)¢ R [since 3(3)# 0]

R is not transitive because (13),(3,9)€R, but (1,9)£ R[3(1)-940]
Hence, R is neither reflexive nor symmetric nor transitive. VR

23. 00%

Q-7 Determine whether each of the following relations are reflexive, symmetric
and transitive:

Relation R in the set A = (1, 2, 3, 4, 5, 6) as
R= {(x, y) : y is divisible by x}

Solution

24. 00%

Q-7 Determine whether each of the following relations are reflexive, symmetric and
transitive:

Relation R in the set A = (1, 2, 3, 4, 5, 6) as

R = ((x y) : y is divisible by x}

Solution: — R=((x,y): y is divisible by x}
We know that any number other than 0 is divisible by itself.
Thus, (x,x) € À
So, R is reflexive.
(2,4) &:R à {because 4 is divisible by 2]
But (4,2) &R [since 2 is not divisible by 4]
So, R is not symmetric.
Let (x,y) and (9,2) € R. So, y is divisible by x and z is divisible by y.
So, z is divisible by x > (x,2) € R

So, R is transitive.
So, R is reflexive and transitive but not symmetric.

25. ag

Q-8 Determine whether each of the following relations are reflexive, symmetric and
transitive:

Relation R in the set Z of all integers defined as
R = {(x, y) : x- y is an integer}

Solution

26. ag

Q-8 Determine whether each of the following relations are reflexive, symmetric and
transitive:
Relation R in the set Z of all integers defined as R = {(x, y) : x—y is an integer}

Solution: — R={(x,y):x—y is an integer)

For x € Z, (x,x) € Rbecause x=x=0 is an integer.

So, R is reflexive.

For, x,» € Z, ify ER, then X-) is an integer > (Y-X)is an integer.
So, (y,x) ER

So, R is symmetric.

Let (x,) and (9,2) € R, where %),2 € Z,

> (1 y) and (Y=2)are integers.

> 1-2=(x- y)+(y-2) is an integer.

So, R is transitive.
So, R is reflexive, symmetric and transitive.

27. Q-9 Determine whether each of the fonowmpgrrerátions are reflexive, symmetric and
transitive:

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}

Solution

28. Solution 1? Ki

a) R= { (3 y) :x and y work at the same place |

Ris reflexive because (4X) € R
R is symmetric because ,

1£ (1,7) €R, then xand Y workat the Same place and Y and x also work at the

same place. (yx)eR,

R is transitive because,

Let (HP). (52) ER

x and Y work atthe same place and Y and 2 work at the same place.

Then, x and 2 also works at the same place. (nz)eR,
Hence, R is reflexive, symmetric and transitive. \R

29. Solution A

b) R= i(x,y):x and y live in the same locality}
R is reflexive because (%*) € À
R is symmetric because,
14 (3,3) €R then xand Y live inthe Same locality and Y and x also live in the
same locality (nr)eR,
R is transitive because,
Let (x,y),.(y,2) € R
x and )-live in the same locality and Y and 2 live in the same locality,

Thenx and 2 also live in the same locality. (nz)ER,

Hence, R is reflexive, symmetric and transitive. VR

30. Solution A

c) R= (x y): x is exactly 7cm taller than y!
Ris not reflexive because (+) ER
R is not symmetric because,
Ir (xy) ER then xis exactly Temtaller than Y and Y is clearly not taller than x
(er,
R is not transitive because,

Let (x,y):(y,2)€R

x is exactly 7cmtaller than Y and } is exactly 7cmtaller than 2.

Then xis exactly l4cmtaller thanZ. (1,2) 4 R
Hence, R is neither reflexive nor symmetric nor transitive. VR

31. 00%

d) R= ¡(x,y) ¿a is wife of y}

Solution

R is not reflexive because (*+*) € R
R is not symmetric because,

Let (y) ER. vis the wife of Y and Vis not the wife of x. (HER,
R is not transitive because,
Let (x,y):(y,2)€ R
xis wife of Y and Y is wife of z, which is not possible.
(1,2) 8R
Hence, R is neither reflexive nor symmetric nor transitive. VR

32. 00%

e) R= x y): x is father ofy|

Solution

R is not reflexive because (4) R
R is not symmetric because,

Let (13) ER. vis the father of Y and ) is not the father of x. (MX) ER,
R is not transitive because,

Let (x). (Je R

x is father of Y andy is father of z, xis not father of z .(1:2)R.

Hence, Rus neither reflexive nor symmetric nor transitive.

33. ag

Q-10 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.

Solution

34. ag

Q-10 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.

Solution
R={(x,y):x and y have same number of pages!
R is reflexive since (%*)€Ras Yand xhave same number of pages.
“Ris reflexive.

(x,y)eR

xand have same number of pages and Y and x have same number of pages (.” xjeR

VR

“ Ris symmetric,

35. N
(x,y)ER(y,z) € R

x and Ÿ have same number of pages, Y and 2 have same number of pages.
Thenx and 2 have same number of pages.

(x,2)eR

“+ Ris transitive.

R is an equivalence relation.

36. 00%

Q-11 Show that the relation R in R defined as R = ((a, b) : a < b), is reflexive and
transitive but not symmetric.

Solution

37. 00%

Q-11 Show that the relation R in R defined as R = ((a, b) : a < b}, is reflexive and
transitive but not symmetric.

Solution

R={(a,b):a <b} (a,b),(b,c)@R
(a,a)eR a Sb andb<e
“+ Ris reflexive. wh <%
(2,4)eR(as2< ay E Placer

«+ Ris transitive.
(4,2) € R (as 4>2)

R is reflexive and transitive but not symmetric.

“. Ris not symmetric.