Exercise miscellaneous

Show that the function f : R → {x ∈ R : - 1 < x < 1} defined by ( ) 1 | | xf x x= + , x ∈ R is one one and onto function.

Show that the function f : R → R given by f (x) = x3 is injective.

Given a non empty set X, consider P(X) which is the set of all subsets of X. Define the relation R in P(X) as follows: For subsets A, B in P(X), ARB if and only if A ⊂ B. Is R an equivalence relation on P(X)? Justify your answer.

Find the number of all onto functions from the set {1, 2, 3,......, n} to itself.

Let A = {- 1, 0, 1, 2}, B = {- 4, - 2, 0, 2} and f, g : A → B be functions defined by f (x) = x2 - x, x ∈ A and 1( ) 2 1, 2g x x = − − x ∈ A. Are f and g equal? Justify your answer . (Hint: One may note that two functions f : A → B and g : A → B such that f (a) = g(a) ∀ a ∈ A, are called equal functions). MATHEMA TICS16

Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is (A) 1 (B) 2 (C) 3 (D) 4

Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is (A) 1 (B) 2 (C) 3 (D) 4