Each of the following defines a relation on N:
(i) x is greater than y, x, y ∈ N
(ii) x + y = 10, x, y ∈ N
(iii) x y is square of an integer x, y ∈ N
(iv) x + 4y = 10 x, y ∈ N.
Determine which of the above relations are reflexive, symmetric and transitive.
(i) Given, x is greater than y; x, y ∈ N
If (x, x) ∈ R, then x > x, which is not true for any x ∈ N.
Thus, R is not reflexive.
Let (x, y) ∈ R
⇒ xRy
⇒ x > y
So, y > x is not true for any x, y ∈ N
Hence, R is not symmetric.
Let xRy and yRz
⇒ x > y and y > z
⇒ x > z
⇒ xRz
Hence, R is transitive.
(ii) x + y = 10; x, y ∈ N
Thus,
R = {(x, y); x + y = 10, x, y ∈ N}
R = {(1, 9), (2, 8), (3, 7), (4, 6), (5, 5), (6, 4), (7, 3), (8, 2), (9, 1)}
It’s clear (1, 1) ∉ R
So, R is not reflexive.
(x, y) ∈ R ⇒ (y, x) ∈ R
Therefore, R is symmetric.
Now (1, 9) ∈ R, (9, 1) ∈ R, but (1, 1) ∉ R
Therefore, R is not transitive.
(iii) Given, xy is square of an integer x, y ∈ N
R = {(x, y) : xy is a square of an integer x, y ∈ N}
It’s clearly (x, x) ∈ R, ∀ x ∈ N
As x2 is square of an integer for any x ∈ N
Thus, R is reflexive.
If (x, y) ∈ R ⇒ (y, x) ∈ R
So, R is symmetric.
Now, if xy is square of an integer and yz is square of an integer.
Then, let xy = m2 and yz = n2 for some m, n ∈ Z
x = m2/y and z = x2/y
xz = m2n2/ y2, which is square of an integer.
Thus, R is transitive.
(iv) x + 4y = 10; x, y ∈ N
R = {(x, y): x + 4y = 10; x, y ∈ N}
R = {(2, 2), (6, 1)}
It’s clearly seen (1, 1) ∉ R
Hence, R is not symmetric.
(x, y) ∈ R ⇒ x + 4y = 10
And (y, z) ∈ R ⇒ y + 4z = 10
⇒ x – 16z = -30
⇒ (x, z) ∉ R
Therefore, R is not transitive.
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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,