2. Let f : x → Y be an invertible function. Show that the inverse of f^–1 is f, i.e.(f^–1)^–1 = f.

Let S = {a, b, c} and T = {1, 2, 3}. Find F^–1 of the following functions F from S to T, if it exists
(i) F = {(a, 3), (b, 2), (c, 1)} (ii) F = {(a, 2), (b, 1), (c, 1)}

Let S = {1, 2, 3}. Determine whether the functions f : S → S defined as below have inverses. Find f^–1, if it exists.

(a) f = {1, 1), (2, 2), (3, 3)}    (b) f = {(1, 2), (2, 1), (3, 1)}
(c) f = {(1,3),(3,2), (2, 1)}

State with reason whether following functions have inverse (i) f : {(1,2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}
(ii) g : {5,6,7,8}  → {1,2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}
(iii) h : {2, 3, 4, 5} → {7,9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)}

287. Show that * :R x R → R given by (a, b) → a + 4 b² is a binary operation.

Let P be the set of all subsets of a given set X.
Show that U:PxP→P given by (A, B) ∴ A ∪ B and ∩ : P x P → P given by (A, B) → A ∩ B are binary operations on the set P.