Let $S$ be the set of all points in a plane and let $R$ be a relation in $S$ defined by $R=\{(A, B): d(A, B)<2$ units $\}$, where $d(A, B)$ is the distance between the points $A$ and $B$.
Show that $\mathrm{R}$ is reflexive and symmetric but not transitive.
Given that, $\forall A, B \in S, R=\{(A, B): d(A, B)<2$ units $\} .$
Now,
$\underline{R}$ is Reflexive if $(A, A) \in \underline{R} \forall \underline{A} \in \underline{S}$
For any $A \in S$, we have
$\mathrm{d}(\mathrm{A}, \mathrm{A})=0$, which is less than 2 units
$\Rightarrow(A, A) \in R$
Thus, $R$ is reflexive.
$\underline{R}$ is Symmetric if $(A, B) \in \underline{R} \Rightarrow \underline{(B, A)} \in \underline{R} \underline{\forall} \underline{A, B} \in \underline{S}$
$(A, B) \in R$
$\Rightarrow d(A, B)<2$ units
$\Rightarrow \mathrm{d}(\mathrm{B}, \mathrm{A})<2$ units
$\Rightarrow(B, A) \in R$
Thus, $R$ is symmetric.
$\underline{\mathrm{R}}$ is Transitive if $(\mathrm{A}, \mathrm{B}) \in \underline{\mathrm{R}}$ and $(\mathrm{B}, \mathrm{C}) \in \underline{\mathrm{R}} \Rightarrow \underline{(\mathrm{A}, \mathrm{C})} \in \underline{\mathrm{R}} \underline{\forall} \underline{\mathrm{A}, \mathrm{B}, \mathrm{C}} \in \underline{\mathrm{S}}$
Consider points $A(0,0), B(1.5,0)$ and $C(3.2,0)$.
$\mathrm{d}(\mathrm{A}, \mathrm{B})=1.5$ units $<2$ units and $\mathrm{d}(\mathrm{B}, \mathrm{C})=1.7$ units $<2$ units
$\mathrm{d}(\mathrm{A}, \mathrm{C})=3.2 \nless 2$
$\Rightarrow(A, B) \in R$ and $(B, C) \in R \Rightarrow(A, C) \notin R$
Thus, $R$ is not transitive.
Thus, $R$ is reflexive, symmetric but not transitive.
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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,