Mark the tick against the correct answer in the following:
Let $a * b=a+a b$ for all $a, b \in Q$. Then,
A. * is not a binary composition
B. $*$ is not commutative
C. $*$ is commutative but not associative
D. $*$ is both commutative and associative
According to the question ,
$\mathrm{Q}=\{\mathrm{a}, \mathrm{b}\}$
$\mathrm{R}=\{(\mathrm{a}, \mathrm{b}): \mathrm{a} * \mathrm{~b}=\mathrm{a}+\mathrm{ab}\}$
Formula
$*$ is commutative if $a * b=b * a$
$*$ is associative if $(\mathrm{a} * \mathrm{~b}) * \mathrm{c}=\mathrm{a} *(\mathrm{~b} * \mathrm{c})$
Check for commutative
Consider, $a * b=a+a b$
And, $b * a=b+b a$
Both equations will not always be true .
Therefore , * is not commutative ……. (1)
Check for associative
Consider, $(a * b) * c=(a+a b) * c=a+a b+(a+a b) c=a+a b+a c+a b c$
And, $a *(b * c)=a *(b+b c)=a+a(b+b c)=a+a b+a b c$
Both the equation are not the same and therefore will not always be true.
Therefore, $*$ is not associative ....... (2)
Now, according to the equations (1), (2)
Correct option will be (B)
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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,