Mark the tick against the correct answer in the following:
Let $S$ be the set of all straight lines in a plane. Let $R$ be a relation on $S$ defined by a $R$ b $\Leftrightarrow$ a $\|$ b. 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 $S=\{x, y, z\}$
And $R=\{(x, x),(y, y),(z, z)\}$
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
Since, $(x, x) \in R,(y, y) \in R,(z, z) \in R$
Therefore, $R$ is reflexive ....... (1)
Check for symmetric
Since,$(x, x) \in R$ and $(x, x) \in R$
$(y, y) \in R$ and $(y, y) \in R$
$(z, z) \in R$ and $(z, z) \in R$
Therefore, $\mathrm{R}$ is symmetric ....... (2)
Check for transitive
Here, $(x, x) \in R$ and $(y, y) \in R$ and $(z, z) \in R$
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,