Show that the relation $R$ defined on the set $A=(1,2,3,4,5)$, given by
$R=\{(a, b):|a-b|$ is even $\}$ is an equivalence relation.
In order to show $R$ is an equivalence relation we need to show $R$ is Reflexive, Symmetric and Transitive.
Given that, $\forall a, b \in A, R=\{(a, b):|a-b|$ is even $\}$.
Now,
$\underline{R}$ is Reflexive if $(a, a) \in \underline{R} \underline{\forall} \underline{a} \in \underline{A}$
For any a $\in A$, we have
$|a-a|=0$, which is even.
$\Rightarrow(a, a) \in R$
Thus, $R$ is reflexive.
$\underline{R}$ is Symmetric if $(a, b) \in \underline{R} \Rightarrow \underline{(b, a)} \in \underline{R} \underline{\forall} \underline{a, b} \in \underline{A}$
$(a, b) \in R$
$\Rightarrow|\mathrm{a}-\mathrm{b}|$ is even.
$\Rightarrow|b-a|$ is even.
$\Rightarrow(b, a) \in R$
Thus, $R$ is symmetric.
$\underline{R}$ is Transitive if $(a, b) \in \underline{R}$ and $(b, c) \in \underline{R} \Rightarrow(a, c) \in \underline{R} \forall \underline{a}, b, c \in \underline{A}$
Let $(a, b) \in R$ and $(b, c) \in R \forall a, b, c \in A$
$\Rightarrow|a-b|$ is even and $|b-c|$ is even
$\Rightarrow(\mathrm{a}$ and $\mathrm{b}$ both are even or both odd) and ( $\mathrm{b}$ and $\mathrm{c}$ both are even or both odd)
Now two cases arise:
Case 1 : when b is even
Let $(a, b) \in R$ and $(b, c) \in R$
$\Rightarrow|a-b|$ is even and $|b-c|$ is even
$\Rightarrow a$ is even and $c$ is even $[\because b$ is even $]$
$\Rightarrow|a-c|$ is even [ $\because$ difference of any two even natural numbers is even]
$\Rightarrow(a, c) \in R$
Case $2:$ when b is odd
Let $(a, b) \in R$ and $(b, c) \in R$
$\Rightarrow|\mathrm{a}-\mathrm{b}|$ is even and $|\mathrm{b}-\mathrm{c}|$ is even
$\Rightarrow \mathrm{a}$ is odd and $\mathrm{c}$ is odd $[\because \mathrm{b}$ is odd $]$
$\Rightarrow|\mathrm{a}-\mathrm{c}|$ is even $[\because$ difference of any two odd
natural numbers is even]
$\Rightarrow(a, c) \in R$
Thus, $R$ is transitive on $Z$.
Since $R$ is reflexive, symmetric and transitive it is an equivalence relation on $Z$.
- 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,
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,