Mark the tick against the correct answer in the following:
Let $Z$ be the set of all integers and let $R$ be a relation on $Z$ defined by a $R b \Leftrightarrow(a-b)$ is divisible by $3 .$ Then, $R$ is
A. reflexive and symmetric but not transitive
B. reflexive and transitive but not symmetric
C. symmetric and transitive but not reflexive
D. an equivalence relation
According to the question,
Given set $Z=\{1,2,3,4 \ldots . .\}$
And $R=\{(a, b): a, b \in Z$ and $(a-b)$ is divisible by 3$\}$
Formula
For a relation $R$ in set $A$
Reflexive
The relation is reflexive if $(a, a) \in R$ for every $a \in A$
Symmetric
The relation is Symmetric if $(a, b) \in R$, then $(b, a) \in R$
Transitive
Relation is Transitive if $(a, b) \in R \&(b, c) \in R$, then $(a, c) \in R$
Equivalence
If the relation is reflexive, symmetric and transitive, it is an equivalence relation.
Check for reflexive
Consider, $(a, a)$
$(\mathrm{a}-\mathrm{a})=0$ which is divisible by 3
$(a, a) \in R$ where $a \in Z$
Therefore , R is reflexive ……. (1)
Check for symmetric
$\therefore(\mathrm{a}-\mathrm{b})$ which is divisible by 3
- $(\mathrm{a}-\mathrm{b})$ which is divisible by 3
(since if 6 is divisible by 3 then $-6$ will also be divisible by 3 )
$\therefore(b-a)$ which is divisible by $3 \Rightarrow(b, a) \in R$
For any $(a, b) \in R ;(b, a) \in R$
Therefore , R is symmetric ……. (2)
Check for transitive
Consider , (a,b) ∈ R and (b,c) ∈ R
$\therefore(a-b)$ which is divisible by 3
and $(b-c)$ which is divisible by 3
$[(a-b)+(b-c)]$ is divisible by 3$]$ (if 6 is divisible by 3 and 9 is divisible by 3 then $6+9$ will also be divisible by 3)
$\therefore(\mathrm{a}-\mathrm{c})$ which is divisible by $3 \Rightarrow(\mathrm{a}, \mathrm{c}) \in \mathrm{R}$
Therefore $(a, b) \in R$ and $(b, c) \in R$ then $(a, c) \in R$
Therefore, $\mathrm{R}$ is transitive ....... (3)
Now, according to the equations (1), (2), (3)
Correct option will be (D)
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,