Let $*$ be a binary operation on $N$, defined by $a * b=a^{b}$ for all $a . b \in N$.
Show that * is neither commutative nor associative.
To prove: $*$ is neither commutative nor associative
Let us assume that $*$ is commutative
$\Rightarrow a^{b}=b^{a}$ for $a l l a, b \in N$
This is valid only for $\mathrm{a}=\mathrm{b}$
For example take $a=1, b=2$
$1^{2}=1$ and $2^{1}=2$
So * is not commutative
Let us assume that * is associative
$\Rightarrow\left(a^{b}\right)^{c}=a^{b^{c}}$ for $a l l a, b, c \in N$
$\Rightarrow a^{b c}=a^{b^{c}}$ for all $a, b, c \in N$
This is valid in the following cases:
(i) $a=1$
(ii) $b=0$
(iii) $\mathrm{bc}=\mathrm{b}^{\mathrm{C}}$
For example, let $\mathrm{a}=2, \mathrm{~b}=1, \mathrm{c}=3$
$a^{b c}=2^{(1 \times 3)}=2^{3}=8$
$a^{b^{c}}=2^{1^{3}}=2$
So * is not associative
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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,