Let R1 and R2 be two relations defined on ℝ by a R1b ⇔ ab ≥ 0 and aR2b ⇔ a ≥ b. Then,
- R1 is an equivalence relation but not R2
- R2 is an equivalence relation but not R1
- Both R1 and R2 are equivalence relations
- Neither R1 nor R2 is an equivalence relation
The Correct Option is D
Solution and Explanation
The correct answer is (D):
aR1b ⇔ ab ≥ 0
So, definitely (a, a) ∈ R1 as a2 ≥ 0
If (a, b) ∈ R1 ⇒ (b, a) ∈ R1
But if (a, b) ∈ R1, (b, c) ∈ R1
⇒ Then (a, c) may or may not belong to R1
{Consider a = –5, b = 0, c = 5 so (a, b) and (b, c) ∈ R1 but ac < 0}
So, R1 is not equivalence relation
a R2 b ⇔ a ≥ b
(a, a) ∈ R2 ⇒ so reflexive relation
If (a, b) ∈ R2 then (b, a) may or may not belong to R2
⇒ So not symmetric
Hence it is not equivalence relation
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Concepts Used:
Types of Differential Equations
There are various types of Differential Equation, such as:
Ordinary Differential Equations:
Ordinary Differential Equations is an equation that indicates the relation of having one independent variable x, and one dependent variable y, along with some of its other derivatives.
\(F(\frac{dy}{dt},y,t) = 0\)
Partial Differential Equations:
A partial differential equation is a type, in which the equation carries many unknown variables with their partial derivatives.
Linear Differential Equations:
It is the linear polynomial equation in which derivatives of different variables exist. Linear Partial Differential Equation derivatives are partial and function is dependent on the variable.
Homogeneous Differential Equations:
When the degree of f(x,y) and g(x,y) is the same, it is known to be a homogeneous differential equation.
\(\frac{dy}{dx} = \frac{a_1x + b_1y + c_1}{a_2x + b_2y + c_2}\)
Read More: Differential Equations