The document discusses relations, functions, and their properties. Some key points: 1. Relations are subsets of the Cartesian product A x B. The number of relations from set A to set B is 2^mn, where m and n are the number of elements in sets A and B respectively. 2. Functions are a special type of relation where each element of the domain is mapped to only one element of the codomain. 3. Logarithmic and exponential functions are introduced along with their graphs and properties. Logarithmic functions are defined only for positive bases and arguments. 4. Modulus functions are discussed through their graphs and the concept of opening modulus. Methods to solve modulus equations are

1. Relations and functions
One Shot

2. Nishant Vora
B.Tech - IIT Patna

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12. CROSS PRODUCT
& RELATIONS

13. Cross Product (A x B)
A = {1, 2, 3}
B = {a, b}
Ordered Pair (x, y) ≠ (y, x)
Number of elements
n(A x B) = n(A).n(B)

14. A x B = {(1, a), (1, b), (2, a) (2, b), (3, a), (3, b)} True or False ?
A. 1 ∈ A
B. 1 ∈ A x B
C. (1, a) ∈ A x B
D. { (1, a) } ⊂ A x B

16. How many subsets will the following sets have?🤔
A = {1, 2} A = { 1, 2, 3 }

17. Conclusion
Number of subsets of set A = 2n
where n → number of element in set A

18. A x B = {(1, a), (1, b), (2, a) (2, b), (3, a), (3, b)}
How many subsets will A x B have ?

19. Let Z be the set of integers. If A = {x∈Z : 2(x+2)(x2 - 5x + 6) =1} and
B = {x∈Z : -3 < 2x - 1 < 9}, then the number of subsets of the set
A x B, is
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A. 215
B. 218
C. 212
D. 210

21. Relations
A subset of A x B
If n(A) = m & n(B) = n then the number
of Relations = 2mn

22. Visual Representation (Arrow Diagram)
1
2
3
a
b
A B

23. Relations in Real life

24. A = { 1, 2, 3}
B = {a , b, c }
R: A ➝ B
R = {(1, a), (2, b)}

25. Identity Relations
Every element of A is related
to itself only

26. Representation of Relations
1. Roster form
2. Set builder form

27. TYPES OF
RELATIONS

28. Types of Relations
Reflexive
(a, a) ∈ R for all a ∈ A
Symmetric
If (a, b) ∈ R then (b, a) ∈ R Transitive
If (a, b) ∈ R and (b, c) ∈ R
then (a, c) ∈ R
Anti- Symmetric
If (a, b) ∈ R and (b, a) ∈ R ⇒ a = b

29. The relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}
on set A = {1, 2, 3} is -
A. Reflexive but not symmetric
B. Reflexive but not transitive
C. Symmetric and transitive
D. Neither symmetric nor transitive

31. The relation R = {(1, 1), (2, 2), (1, 2), (2, 3)} on set
A = {1, 2, 3} is -
A. Reflexive but not symmetric
B. Reflexive but not transitive
C. Symmetric and transitive
D. Neither symmetric nor transitive

33. Let R = {(3, 3), (6, 6), (9, 9), (12, 12) (6, 12) (3, 9) (3, 12), (3, 6)} be a
relation on the set A = {3, 6, 9, 12}. The relation is -
A. An equivalence relation
B. Reflexive and symmetric only
C. Reflexive and transitive only
D. Reflexive only
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35. ह ाँ - न bol

36. R = { (a, b): a is divisible by b}

37. R = {(T1, T2) : T1 ≅ T2}

38. R = {(a, b) : a is brother of b}

39. R = {(a, b) : a is cousin of b}

40. For α ∊ N , consider a relation R on N given by R = {(x, y) : 3x + αy is
a multiple of 7}. The relation R is an equivalence relation if and
only if :
A. α = 14
B. α is a multiple of 4
C. 4 is the remainder when α is divided by 10
D. 4 is the remainder when α is divided by 7
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42. Let R1 and R2 be two relations defined on R by
aR1b ⇔ ab ≥ 0 and aR2b ⇔ a ≥ b, then
A. R1 is an equivalence relation but not R2
B. R2 is an equivalence relation but not R1
C. Both R1 and R2 are equivalence relations
D. Neither R1 nor R2 is an equivalence relation
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44. Let R1 = {(a, b) ∈ N x N : |a - b| ≤ 13} and
R2 = {(a, b) ∈ N x N : |a - b| ≠ 13}. Then on N:
A. Both R1 and R2 are equivalence relations
B. Neither R1 nor R2 is an equivalence relation
C. R1 is an equivalence relation but R2 is not
D. R2 is an equivalence relation but R1 is not
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46. Let R be relation from the set {1, 2, 3 ….., 60} to itself such that
R = {(a, b) : b = pq, where p, q ≥ 3 are prime numbers }. Then, the
number of elements in R is :
A. 600
B. 660
C. 540
D. 720
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48. Which of the following is not correct for relation R on the set of
real numbers ?
A. (x, y) ∊ R ⇔ 0 < |x| - |y| ≤ 1 is neither transitive nor symmetric
B. (x, y) ∈ R ⇔ 0 < |x - y| ≤ 1 is symmetric and transitive.
C. (x, y) ∈ R ⇔ |x| - |y| ≤ 1 is reflexive but not symmetric
D. (x, y) ∈ R ⇔ |x - y| ≤ 1 is reflexive and symmetric.
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50. Let N be the set of natural numbers and a relation R on N be
defined by R = {(x,y) ∈ N x N : x3 - 3x2y - xy2 + 3y3 = 0}. Then the
relation R is
A. Symmetric but neither reflexive nor transitive.
B. Reflexive but neither symmetric nor transitive.
C. Reflexive and symmetric, but not transitive.
D. An equivalence relation.
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52. Summary
● Relations are subset of A x B
● Number of subsets of A x B = Number of relations defined from
A → B
● Number of subsets = 2n
● n(AxB) = n(A).n(B)
● Number of relation which can be defined from A → B is 2mn
● Relations can be represented in roaster as well as set builder
form
● Arrow starts from anywhere and ends anywhere (Relations are
complicated)

53. DOMAIN,
CODOMAIN AND
RANGE

54. Domain, Codomain and Range of Relations
A = {1, 3, 5, 7}; B = {2, 4, 6, 8}
R = {(3, 2), (5, 4), (7, 2), (7, 4), (7, 6)}
Domain =
Range =
Codomain =
A B
3
5
7
2
4
6
1
8

55. Domain, Codomain and Range of Relations
Domain of R : Collection of all elements of A which has a image in B
Range of R : Collection of all elements of B which has a pre-image in A

56. If R = {(x, y)| x, y ∈ Z, x2 + y2 ≤ 4} is a relation in Z, then domain
of R is -
A. {0, 1, 2}
B. {0, -1, -2}
C. {-2, -1, 0, 1, 2}
D. None of these
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58. Equivalence Class
Let R = { (1, 1), (2, 2), (3, 3), (1, 2), (2, 1) }
Equivalence class of 1 = [1] =
Equivalence class of 2 = [2] =
Equivalence class of 3 = [3] =

59. If R = {P, Q) | P and Q are at the same distance from the origin }
be a relation, then the equivalence class of (1, -1) is the set :
26 Feb 2021 Shift 1
A. S = { (x,y) | x2 + y2 = 1 }
B. S = { (x,y) | x2 + y2 = 4}
C. S = {(x,y) | x2 + y2 = √2 }
D. S = { (x,y) | x2 + y2 = 2 }

61. INVERSE OF
RELATION

62. Inverse of a Relation
A = { 1, 2, 3, 4 }
B = { a, b, c, d }
R : A → B
R = {(1, a), (2, b), (3, c), (4, d)}

63. 1
2
3
4
a
b
c
d
B A
a
b
c
d
1
2
3
4
A B

64. Domain of R = Range of R-1
Range of R = domain of R-1

65. If R = {(x, y); x, y ∈ Z, x2 + 3y2 ≤ 8} is a relation on the set of integers
Z, then the domain of R-1 is :
A. {0, 1}
B. {-2, -1, 1, 2}
C. {-1, 0, 1}
D. {-2, -1, 0, 1, 2}
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68. Functions

69. Functions
Modulus
1
Logarithmic
2
Exponentials
3
Greatest integer functions
4
Fractional part
5

70. Modulus Functions

71. Modulus Functions: Definition
|x| |x|

72. Modulus Functions: Domain, Range, Graphs

73. Opening Modulus Functions

74. Opening Modulus Functions

75. Opening Modulus Functions
On the right of critical point modulus opens with a plus sign
and on the left it opens with a minus sign.
NOTE -

76. Graph of Modulus
Function

77. Graph of Modulus Functions
|x| + k or |x| - k

78. Graph of Modulus Functions
|x + k| or |x - k|

79. Graph of Modulus Functions
k|x| or |kx|

80. Define f(x) free of modulus and draw the graph f(x) = |2x - 3|

82. Define f(x) free of modulus and draw the graph f(x) = |2x| - 3

84. Double Mod Graphs

85. f(x) = |x - 1| + |x - 2| Shortcut

87. f(x) = | x | + |x - 1| Shortcut

89. f(x) = |x + 1| + |x - 1| Shortcut

91. f(x) = |2x + 1| + |3x - 1| Shortcut

93. Open modulus and draw graph of f(x) = |x + 1| + |x| + |x - 1|

95. Find ‘k’ for which following equation has 4 solutions
|x - 1| - 2|x| + |2x + 1| = k

98. Graphical Transformation

99. Graphical Transformation
y = f(x) y = -f(x)
Flip in x axis

100. Graphical Transformation
y = f(x) y = f(-x)
Flip in y axis

101. Graphical Transformation
y = f(x) y = |f(x)|
neeche wala uper

102. Graphical Transformation
y = f(x) y = f(|x|)
Right side = Same
Left side = Mirror

103. Graphical Transformation
y = f(x) |y| = f(x)
Upar wala = Same
Neeche = Mirror

104. Draw the graph of y = ||x - 1| - 2|

106. Draw the graph of y = -|x + 2|

107. Draw the graph of |y| + x = -1

108. Modulus Equations

109. Solving Modulus Equations
|f(x)| = a ⇒ f(x) = ± a

110. Solving Modulus Equations (Method - 1)
Ex. |x - 1| = 3

111. Solving Modulus Equations (Method - 2)
Ex. |x - 1| = 3

112. Solving Modulus Equations
Ex. ||x| - 1| = 7

113. Solve for x : ||x + 1| - 4| = 5

114. Solve for x : |x - 1| - |x| = 1 M-1: Algebraic Method

116. Solve for x : |x - 1| - |x| = 1 M-2: Graphical Method

118. Solve the following linear equation ||x - 1| - 2| = |x - 3|
A. x ∈ [1, ∞) B. x ∈ [3, ∞)
C. x = 3 D. x = 1

120. Number of solutions of the equations |2x2 + x - 1| = |x2 + 4x + 1|
A. 1 B. 2 C. 3 D. 4

122. Solutions of |4x + 3| + |3x - 4| = 12 are
A. B.
C. D.

126. Log Function

127. Log Functions : Definitions

128. Find value of:
i. log81 27 iii. log1/3 9√3
ii. log10 100

129. Domain: Restriction on base and input
loga x is defined if a > 0, a ≠ 1 and x > 0

130. Frequently used base values
If a = 10, then write log N rather than log10 N
If a = e, we write lnN rather than loge N

131. Find the value of
i. log(2 - √3) (2 + √3) = k
ii. log(1/3) (3√3) = k

133. Graph of Log Function

134. Graphs of Log Functions
y = log2 x
x
y

135. Graphs of Log Functions
y = log1/2 x
x
y

136. Graphs of Log Functions
y = loga x

137. Graphs of Log Functions
y = log2 (x + k) or y = log2 (x - k)

138. Graphs of Log Functions
y = log2 (x) + k or y = log2 (x) - k

139. Graphs of Log Functions
y = k log2 (x)

140. Properties of Log
Functions

141. Properties of Log Functions
1 loga 1 = 0
2 loga a = 1
3 loga (mn) = logam+ logan
4 loga (m/n) = logam- logan

142. Properties of Log Functions
a loga x = x
5
6
7
8

143. Solving Log Equations

144. Solving Log Equations

145. (log2 3).(log3 4).(log4 5) … logn(n + 1) = 10. Find n = ?
A. 512 B. 128 C. 1024 D. 1023

147. Solve
A. 0 B. 1 C. 2 D. 2

149. Solve
A. 0 B. 1 C. -2 D. -2, 1

150. Find the value of x :

152. Find the value of x :

154. [ JEE-Advanced 2018 ]

155. [ JEE-Advanced 2013, 4, (-1) ]

157. A. 2f(x)
B. 2f(x2)
C. (f(x))2
D. -2f(x)
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158. Greatest Integer
Function

159. Greatest Integer Function (G.I.F)
Definition: Greatest integer less than or equal to x.

161. Graph of G.I.F

162. Graph of G.I.F

163. Domain and Range of G.I.F.
Domain:
Range:

164. Properties of G.I.F
1. x-1 < [x] ≤ x
1. [x + n] = [x] if n is integer
1. [x + [x]] = [x]

165. Properties of G.I.F

166. Fractional Part Function

167. Fraction Part Function (F.P.F)
Definition:

169. Graph of Fractional Part Function

170. Graph of Fractional Part Function

171. Domain and Range of {x}
Domain:
Range:

172. Properties of {x}
1. 0 ≤ {x} < 1
1. x = [x] + {x}
1. {x + n} = {x} if n is integer

173. Properties of {x}

174. Solve the equation
4[x] = x + {x}

176. The number of solution(s) of the equation [2x] – 3 {2x} = 1, is / are

178. The number of solution(s) of the equation [x] + {–x} = 2x, is / are

180. If [x] be the greatest integer less than or equal to x, then
is equal to.
A. 0
B. 4
C. -2
D. 2
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184. Exponential Function

185. Exponential Functions

186. Graph of Exponential Functions

188. Exponential Functions
Domain:
Range:

189. Signum Function

190. Signum Function
or

191. Graph of Signum Function

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