A binary operation $*$ on the set $(0,1,2,3,4,5)$ is defined as
$a * b= \begin{cases}a+b ; & \text { if } a+b<6 \\ a+b-6 ; & \text { if } a+b \geq 6\end{cases}$
Show that 0 is the identity for this operation and each element a has an inverse (6 - a)
To find: identity and inverse element
For a binary operation if $a * e=a$, then e $s$ called the right identity
If $\mathrm{e}^{*} \mathrm{a}=\mathrm{a}$ then $\mathrm{e}$ is called the left identity
For the given binary operation,
$e^{*} b=b$
⇒ e + b = b
⇒ e = 0 which is less than 6.
b*e = b
⇒ b + e = b
⇒ e = 0 which is less than 6
For the $2^{\text {nd }}$ condition,
e*b = b
$\Rightarrow \mathrm{e}+\mathrm{b}-6=\mathrm{b}$
$\Rightarrow \mathrm{e}=6$
But $e=6$ does not belong to the given set $(0,1,2,3,4,5)$
So the identity element is 0
An element $c$ is said to be the inverse of $a$, if $a * c=e$ where $e$ is the identity element (in our case it is $0)$
$a^{*} c=e$
$\Rightarrow \mathrm{a}+\mathrm{c}=\mathrm{e}$
$\Rightarrow \mathrm{a}+\mathrm{c}=0$
$\Rightarrow \mathrm{c}=-\mathrm{a}$
a belongs to $(0,1,2,3,4,5)$
- a belongs to $(0,-1,-2,-3,-4,-5)$
So $c$ belongs to $(0,-1,-2,-3,-4,-5)$
So $c=-a$ is not the inverse for all elements a
Putting in the $2^{\text {nd }}$ condition
$a^{*} c=e$
$\Rightarrow a+c-6=0$
$\Rightarrow c=6-a$
$0 \leq a<6$
$\Rightarrow-6 \leq-a<0 \Rightarrow 0 \leq 6-a<60 \leq c<5$
So $c$ belongs to the given set
Hence the inverse of the element $\mathrm{a}$ is $(6-\mathrm{a})$
Hence proved
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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,