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.
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.
Check whether the relation R in the set R of real numbers, defined by R = {(a, b) : 1 + ab > 0}, is reflexive, symmetric or transitive.
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.
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 ?
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 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.
Find the particular solution of the following differential equation given that at x = 2, y = 1
Find the particular solution of the differential equation : 4x cosec x, (x ≠ 0), given, that y = 0,when
Find the equation of tangents to the curve y = x³ + 2x – 4 which are perpendicular to the line x + 14y – 3 = 0.
Find the angle of intersection of the curves x² + y² = 4 and (x – 2)² + y²= 4, at the point in the first quadrant.
Find the point on the curve y = x³ – 11x + 5 at which the equation of tangent is y = x – 11.