1. Let f : R R be defined as f (x) = 10x + 7. Find the function g : R R such that g o f = f o g = 1R. 2. Let f : W W be defined as f (n) = n 1, if n is odd and f (n) = n + 1, if n is even. Show that f is invertible. Find the inverse of f. Here, W is the set of all whole numbers. 3. If f : R R is defined by f(x) = x2 3x + 2, find f (f (x)). 4. Show that the function f : R {x R : 1 < x < 1} defined by ( ) 1 | | x f x x = + , x R is one one and onto function. 5. Show that the function f : R R given by f (x) = x3 is injective. 6. Give examples of two functions f : N Z and g : Z Z such that g o f is injective but g is not injective. (Hint : Consider f (x) = x and g(x) = |x|). 7. Give examples of two functions f : N N and g : N N such that g o f is onto but f is not onto. (H
Question 1:Let f : R → R be defined as f (x) = 10x + 7. Find the function g : R → R such that g o f = f o g = 1R.
Question 2:Let f : W → W be defined as f (n) = n – 1, if n is odd and f (n) = n + 1, if n is even. Show that f is invertible. Find the inverse of f. Here, W is the set of all whole numbers.
Question 3:If f : R → R is defined by f(x) = x2 – 3x + 2, find f (f (x)).
Question 4:Show that the function f : R → {x Î R : – 1 < x < 1} defined by ( ) 1 | | x f x x = + , x Î R is one one and onto function.
Question 5:Show that the function f : R → R given by f (x) = x3 is injective.
Question 6:Give examples of two functions f : N → Z and g : Z → Z such that g o f is injective but g is not injective. (Hint : Consider f (x) = x and g(x) = |x|).
Question 7:Give examples of two functions f : N → N and g : N → N such that g o f is onto but f is not onto. (Hint : Consider f (x) = x + 1 and 1if 1 ( ) 1 if 1 x x g x x − > = =
Question 8: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 Ì B. Is R an equivalence relation on P(X)? Justify your answer.
Question 9:Given a non-empty set X, consider the binary operation * : P(X) × P(X) → P(X) given by A * B = A Ç B " A, B in P(X), where P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation *.
Question 10:Find the number of all onto functions from the set {1, 2, 3, ... , n} to itself.
Question 11: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)}
Question 12:Consider the binary operations * : R × R → R and o : R × R → R defined as a *b = |a – b| and a o b = a, " a, b Î R. Show that * is commutative but not associative, o is associative but not commutative. Further, show that "a, b, c Î R, a * (b o c) = (a * b) o (a * c). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.
Question 13:Given a non-empty set X, let * : P(X) × P(X) → P(X) be defined as A * B = (A – B) È (B – A), "A, B Î P(X). Show that the empty set f is the identity for the operation * and all the elements A of P(X) are invertible with A–1 = A. (Hint : (A – f) È (f – A) = A and (A – A) È (A – A) = A * A = f).
Question 14:Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as , if 6 6 if 6 a b a b a b a b a b + + < * = + − + ³ Show that zero is the identity for this operation and each element a ¹ 0 of the set is invertible with 6 – a being the inverse of a.
Question 15: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 Î A and 1 ( ) 2 1, 2 g x = x − − x Î 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 Î A, are called equal functions).
Question 16: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
Question 17:Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is (A) 1 (B) 2 (C) 3 (D) 4
Question 18:Let f : R → R be the Signum Function defined as 1, 0 ( ) 0, 0 1, 0 x f x x x > = = − < and g : R → R be the Greatest Integer Function given by g(x) = [x], where [x] is greatest integer less than or equal to x. Then, does fog and gof coincide in (0, 1]?
Question 19:Number of binary operations on the set {a, b} are (A) 10 (B) 16 (C) 20 (D ) 8
