Let $\mathrm{S}$ be the set of all real numbers and let
$\mathrm{R}=\{(\mathrm{a}, \mathrm{b}): \mathrm{a}, \mathrm{b} \in \mathrm{S}$ and $\mathrm{a}=\pm \mathrm{b}\}$
Show that $R$ is an equivalence relation on $S$.
In order to show $R$ is an equivalence relation we need to show $R$ is Reflexive, Symmetric and Transitive.
Given that, $\forall a, b \in S, R=\{(a, b): a=\pm b\}$
Now,
$\underline{R}$ is Reflexive if $(a, a) \in \underline{R} \underline{\forall} \underline{a} \in \underline{S}$
For any $a \in S$, we have
$a=\pm a$
$\Rightarrow(a, a) \in R$
Thus, $R$ is reflexive.
$\underline{R}$ is Symmetric if $(a, b) \in \underline{R} \Rightarrow(b, a) \in \underline{R} \forall \underline{a}, b \in \underline{S}$
$(a, b) \in R$
$\Rightarrow a=\pm b$
$\Rightarrow b=\pm a$
$\Rightarrow(b, a) \in R$
Thus, $R$ is symmetric.
$\underline{R}$ is Transitive if $(a, b) \in \underline{R}$ and $(b, c) \in \underline{R} \Rightarrow(a, c) \in \underline{R} \underline{\forall} a, b, c \in \underline{S}$
Let $(a, b) \in R$ and $(b, c) \in R \forall a, b, c \in S$
$\Rightarrow \mathrm{a}=\pm \mathrm{b}$ and $\mathrm{b}=\pm \mathrm{c}$
$\Rightarrow \mathrm{a}=\pm \mathrm{c}$
$\Rightarrow(\mathrm{a}, \mathrm{c}) \in \mathrm{R}$
Thus, $R$ is transitive.
Hence, $R$ is an equivalence relation.
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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,