Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad(b + c) = bc(a + d). Show that R is an equivalence relation.
Show that the relation R in the Set A = {1, 2, 3, 4,5} given by R = {(a, b) : |a – b| is divisible by 2} is an equivalence relation. Write all the equivalence classes of R.
Show that the relation R on the set Z of all integers defined by (x, y) ∈ is divisible by 3 is an equivalence relation.
Show that the relation R in the set N × N defined by (a, b) R (c, d) if a² + d² = b² + c² is an equivalence relation.
If Z is the set of all integers and R is the relation on Z defined as R = {(a, b) : a, b ∈ Z and a – b is divisible by 5}. Prove that R is an equivalence relation.
Show that the relation S in the set R of real numbers defined as S = {(a, b) : a, b ∈ R and a ≤ b³} is neither reflexive nor symmetric nor transitive.
Let f : X -> Y be a function. Define a relation R on X given be R = {(a, b) : f(a) = f(b)}. Show that R is an equivalence relation ?
Can be a solution of the following differential equation ? If no, find the solution of the D.E.
Find the points on the curve x² + y² – 2x – 3 = 0 at which tangent is parallel to x-axis.
A and B throw a pair of dice alternatively, till one of them gets a total of 10 and wins the game. Find their respective probabilities of winning, if A starts first.
Find the general solution of the differential equation (1 + tan y)(dx – dy) + 2xdy = 0.
Show that the equation of tangent to the parabola y² = 4ax at (x₁, y₁) is yy₁ = 2a(x + x₁).