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.
State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive.
Let R be the equivalence relation in the set A = {0, 1, 2, 3, 4, 5} given by R = {(a, b) : 2 divides (a –b)}. Write the equivalence class [0].
State the reason why the Relation R = {(a, b) : a ≤ b²} on the set R of real numbers is not reflexive.
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 a function f : R → R given by f(x) =ax + b,a, b ∈ R, a ≠ 0 is a bijective.
How many equivalence relations on the set {1, 2, 3} containing (1, 2) and (2, 1) are there in all ?Justify your answer.
If the radius of sphere is measured as 9 cm with an error of 0.03 cm, then find the approximate error in calculating its surface area.
Prove that x² – y² = c(x² + y²)² is the general solution of the differential equation (x² – 3xy²) dx = (y³ – 3x²y) dy, where C is a parameter.
Let f, g : R → R be two functions defined as f (x) = |x| + x and g (x) = |x| – x for all x ∈ R.Then find fog and gof.
Find the particular solution of the differential equation x (1 + y²) dx – y (1 + x²) dy = 0, given that y = 1, when x =0.