Let $A$ be the set of all points in a plane and let $O$ be the origin. Show that the relation $R=\{(P, Q): P, Q \in A$ and $O P=O Q$ ) 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, $A$ be the set of all points in a plane and $O$ be the origin. Then, $R=\{(P, Q): P, Q \in A$ and $O P=$ OQ)\}
Now,
$\underline{R}$ is Reflexive if $(P, P) \in \underline{R} \underline{\forall} \underline{P} \in \underline{A}$
$\forall \mathrm{P} \in \mathrm{A}$, we have
$\mathrm{OP}=\mathrm{OP}$
$\Rightarrow(P, P) \in R$
Thus, $R$ is reflexive.
$\underline{R}$ is Symmetric if $(P, Q) \in \underline{R} \Rightarrow(Q, P) \in \underline{R} \underline{\forall} \underline{P}, Q \in \underline{A}$
Let $P, Q \in A$ such that,
$(P, Q) \in R$
$\Rightarrow \mathrm{OP}=\mathrm{OQ}$
$\Rightarrow \mathrm{OQ}=\mathrm{OP}$
$\Rightarrow(\mathrm{Q}, \mathrm{P}) \in \mathrm{R}$
Thus, $R$ is symmetric.
$\underline{R}$ is Transitive if $(P, Q) \in \underline{R}$ and $(Q, S) \in \underline{R} \Rightarrow(P, S) \in \underline{R} \forall P, Q, S \in A$
Let $(P, Q) \in R$ and $(Q, S) \in R \forall P, Q, S \in A$
$\Rightarrow \mathrm{OP}=\mathrm{OQ}$ and $\mathrm{OQ}=\mathrm{OS}$
$\Rightarrow \mathrm{OP}=\mathrm{OS}$
$\Rightarrow(\mathrm{P}, \mathrm{S}) \in \mathrm{R}$
Thus, $R$ is transitive.
Since $R$ is reflexive, symmetric and transitive it is an equivalence relation on $A$.
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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,