Question:
Let A, B and C be sets. Then show that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Solution:
According to the question,
A, B and C are three given sets
To prove: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Let x ∈ A ∩ (B ∪ C)
⇒ x ∈ A and x ∈ (B ∪ C)
⇒ x ∈ A and (x ∈ B or x ∈ C)
⇒ (x ∈ A and x ∈ B) or (x ∈ A and x ∈ C)
⇒ x ∈ A ∩ B or x ∈ A ∩ C
⇒ x ∈ (A ∩ B) ∪ ( A ∩ C)
⇒ A ∩ (B ∪ C) ⊂ (A ∩ B) ∪ ( A ∩ C) …(i)
Let y ∈ (A ∩ B) ∪ (A ∩ C)
⇒ y ∈ A ∩ B or x ∈ A ∩ C
⇒ (y ∈ A and y ∈ B) or (y ∈ A and y ∈ C)
⇒ y ∈ A and (y ∈ B or y ∈ C)
⇒ y ∈ A and y ∈ (B ∪ C)
⇒ y ∈ A ∩ (B ∪ C)
⇒ (A ∩ B) ∪ (A ∩ C) ⊂ A ∩ (B ∪ C) …(ii)
We know that:
P ⊂ Q and Q ⊂ P ⇒ P = Q
From equations (i) and (ii), we have,
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
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,
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