Let A = (1, -1, i, -i) be the set of four 4th roots of unity. Prepare the composition table for multiplication on A and show that
(i) $A$ is closed for multiplication,
(ii) multiplication is associative on $A$,
(iii) multiplication is commutative on $A$,
(iv) 1 is the multiplicative identity,
(v) every element in A has its multiplicative inverse.
(i) A is said to be closed on * if all the elements of a*b ∈A. composition table is
$\left(\operatorname{as} i^{2}=-1\right)$
As table contains all elements from set A, A is close for multiplication operation.
(ii) For associative, $a \times(b \times c)=(a \times b) \times c$
$1 \times(-i \times i)=1 \times 1=1$
$(1 \times-i) \times i=-i \times i=1$
$a \times(b \times c)=(a \times b) \times c$, so $A$ is associative for multiplication.
(iii) For commutative, $a \times b=b \times a$
$1 \times-1=-1$
$-1 \times 1=-1$
$a \times b=b \times a$, so $A$ is commutative for multiplication.
(iv) For multiplicative identity element $e, a \times e=e \times a=a$ where $a \in A$.
a× e = a
a(e-1) = 0
either a = 0 or e = 1 as a≠0 hence e = 1
So, multiplicative identity element e = 1.
(v) For multiplicative inverse of every element of $A, a * b=e$ where $a, b \in A$.
$1 \times \mathrm{b}_{1}=1$
$b_{1}=1$
$-1 \times b_{2}=1$
$b_{2}=-1$
$\mathrm{i} \times \mathrm{b}_{3}=1$
$\mathrm{b}_{3}=\frac{1}{\mathrm{i}} \Rightarrow \frac{1}{\mathrm{i}} \times \frac{\mathrm{i}}{\mathrm{i}}=\frac{\mathrm{i}}{\mathrm{i}^{2}} \Rightarrow \frac{\mathrm{i}}{-1}=-\mathrm{i}$
$-\mathrm{i} \times \mathrm{b}_{4}=1$
$\mathrm{b}_{4}=\frac{1}{-\mathrm{i}} \Rightarrow \frac{1}{-\mathrm{i}} \times \frac{\mathrm{i}}{\mathrm{i}}=\frac{\mathrm{i}}{-\mathrm{i}^{2}} \Rightarrow \frac{\mathrm{i}}{-(-1)}=\mathrm{i}$
So, multiplicative inverse of $A=\{1,-1,-i, i\}$
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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,