Show that the relation R on the set Z of all integers defined by (x, y) ∈ is divisible by 3 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.
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.
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 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.
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.
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 ?
Three schools A, B and C organised a fete (mela) collecting funds for flood victims in which they sold hand-helds fans, mats and toys made from recycled material, the sale price of each being ₹ 25, ₹ 100 and ₹ 50 respectively. The following table shows the number of articles of each type sold :
Find the angle of intersection of the curves x² + y² = 4 and (x – 2)² + y²= 4, at the point in the first quadrant.
Prove that the function f : [0, ∞) → R given by f(x) = 9x² + 6x – 5 is not invertible. Modify the Codomain of the function f to make it invertible, and hence find f⁻¹.
Find the approximate value of f(3·02), up to 2 places of decimal, where f(x) = 3x² + 5x + 3.