For any two sets A and B, show that the following statements are equivalent:
(i) $A \subset B$
(ii) $A-B=\phi$
(iii) $A \cup B=B$
(iv) $A \cap B=A$.
We have that the following statements are equivalent:
(i) $A \subset B$
(ii) $A-B=\phi$
(iii) $A \cup B=B$
(iv) $A \cap B=A$.
Proof:
Let $A \subset B$
Let $x$ be an arbitary element of $(A-B)$.
NOW,
$x \in(A-B)$
$\Rightarrow x \in A \& x \notin B \quad$ (Which is contradictory)
Also,
$\because A \subset B$
$\Rightarrow A-B \subseteq \phi$ ...(1)
We know that null sets are the subsets of every set.
$\therefore \phi \subseteq A-B$ ...(2)
From $(1) \&(2)$, we get,
$(A-B)=\phi$
$\therefore(\mathrm{i})=(\mathrm{ii})$
Now, We have,
$(A-B)=\phi$
That means that there is no element in $A$ that does not belong to $B$.
Now,
$A \cup B=B$
$\therefore(\mathrm{ii})=(\mathrm{iii})$
WE have,
$A \cup B=B$
$\Rightarrow A \subset B$
$\Rightarrow A \cap B=A$
$\therefore($ iii $)=($ iv $)$
WE have,
$A \cap B=A$
It should be possible if $A \subset B$.
Now,
$A \subset B$
$\therefore(\mathrm{iv})=(\mathrm{i})$
We have,
$(\mathrm{i})=(\mathrm{ii})=(\mathrm{iii})=(\mathrm{iv})$
Therefore, we can say that all statements are equivalent.
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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,