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.
Let A = {1, 2, 3, 4}. Let R be the equivalence relation on A × A defined by (a, b)R(c, d) if a + d = b + c. Find the equivalence class [(1,3)].
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].
Let A = {1, 2, 3, . ...., 9} and R be the relation in A × A defined by (a, b) R (c, d) if a + d =b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation. Also obtain the equivalence class [(2, 5)].
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 ?
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.
Let f : N → R be a function defined as f(x) = 4x² + 12x + 15. Then show that f : N → S, where S is range of f, is invertible. Also find the inverse of f.
Consider given by f(x) = Show that f is bijective. Find the inverse of f and hence find f⁻¹(0) and x such that f⁻¹(x) = 2.
Find the equation of the normal at a point on the curve x² = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.
Find the particular solution of the differential equation (x – sin y) dy + (tan y) dx = 0, given that y = 0 when x = 0.
Find the equation of tangent and normal to the curve x = 1 – cos θ, y = θ – sin θ at θ =
Consider given by f(x) = 5x² + 6x – 9.Prove that f is invertible with f⁻¹(y) = [where, R⁺ is the set of all nonnegative real numbers.]
Consider the experiment of tossing a coin. If the coin shows head, toss is done again, but if it shows tail, then throw a die. Find the conditional probability of the events that ‘the die shows a number greater than 4’, given that ‘there is atleast one tail’.
Find the equation of the tangent line to the curve y =x² -2x + 7 which is
(i) parallel to the line 2x-y +9 =0
(ii) perpendicular to the line 5y - 15x = 13.