Let f : R → R be the function defined by . Then, find the range of f.

Let X = {1, 2, 3} and Y = {4, 5}. Find whether the following subsets of X × Y are functions from X to Y or not.

(i) f = {(1, 4), (1, 5), (2, 4), (3, 5)}

(ii) g = {(1, 4), (2, 4), (3, 4)}

(iii) h = {(1,4), (2, 5), (3, 5)}

(iv) k = {(1,4), (2, 5)}.

If functions f : A → B and g : B → A satisfy g o f = I_A, then show that f is one one and g is onto.

Let n be a fixed positive integer. Define a relation R in Z as follows: ∀ a, b ∈Z, aRb if and only if a – b is divisible by n. Show that R is an equivalence relation.

If A = {1, 2, 3, 4 }, define relations on A which have properties of being:

(a) reflexive, transitive but not symmetric

(b) symmetric but neither reflexive nor transitive

(c) reflexive, symmetric and transitive.