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.
LetA = {x ∈ Z:0≤ x ≤12}. Show that R={(a,b):a,b ∈ A, |a-b| is divisible by 4} is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2].
If f : R → R defined as f(x) = is an invertible function, write
If A = {1, 2, 3}, B = {4, 5, 6, 7} and f = {(1, 4), (2, 5),(3, 6)} is a function from A to B. State whether f is one-one or not.
If f is an invertible function, defined as f(x) = , then write
What is the range of the function f(x) = , x ≠ 1?
If the function f : R → R defined by f(x) = 3x - 4 is invertible, then find
If f : R→ R defined by f(x) = is an invertible function, then find
Let A = R – {3}, B = R – {1}. Let f : A → B be defined by f(x) = Show that f is bijective. Also, find
(i) x, if = 4
(ii)
Let f : W → W be defined as show that f is invertible. Find the inverse of f,where W is the set of all whole numbers.
Let A = R – {2}, B = R – {1}. If f : A → B is a function defined by show that f is one-one and onto. Hence find
Show that the function f in A = defined as f(x) = is one-one and onto. Hence find
Consider f : given by f(x) = x² + 4 Show that f is invertible with the inverse of f given by = , where is the set of all nonnegative real numbers.
Let f : N → N be defined as
for all n ∈ N. State whether the function f is
bijective. Justify your answer.