Mark the tick against the correct answer in the following:
Let $S$ be the set of all real numbers and let $R$ be a relation on $S$ defined by $a R b \Leftrightarrow a^{2}+b^{2}=1 .$ Then, $R$ is
A. symmetric but neither reflexive nor transitive
B. reflexive but neither symmetric nor transitive
C. transitive but neither reflexive nor symmetric
D. none of these
According to the question,
Given set $S=\{\ldots \ldots,-2,-1,0,1,2 \ldots \ldots\}$
And $R=\left\{(a, b): a, b \in S\right.$ and $\left.a^{2}+b^{2}=1\right\}$
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
Consider, $(a, a)$
$\therefore a^{2}+a^{2}=1$ which is not always true
Ex_if $a=2$
$\therefore 2^{2}+2^{2}=1 \Rightarrow 4+4=1$ which is false.
Therefore , R is not reflexive ……. (1)
Check for symmetric
$a R b \Rightarrow a^{2}+b^{2}=1$
b $R a \Rightarrow b^{2}+a^{2}=1$
Both the equation are the same and therefore will always be true.
Therefore , R is symmetric ……. (2)
Check for transitive
$a R b \Rightarrow a^{2}+b^{2}=1$
b $\mathrm{R} \mathrm{c} \Rightarrow \mathrm{b}^{2}+\mathrm{c}^{2}=1$
$\therefore a^{2}+c^{2}=1$ will not always be true
Ex $_{-} \mathrm{a}=-1, \mathrm{~b}=0$ and $\mathrm{c}=1$
$\therefore(-1)^{2}+0^{2}=1,0^{2}+1^{2}=1$ are true
But $(-1)^{2}+1^{2}=1$ is false.
Therefore , R is not transitive ……. (3)
Now, according to the equations (1), (2), (3)
Correct option will be (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,