Chapter 1 Relations and Functions
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- Let R → R be defined as f(x) = 10x + 7. Find the function g : R → R such that gof = fog = IR. Let f : X → Y, where X ⊆ R, Y ⊆ R. Consider an arbitrary element y of Y. By definition, y = 10x + 7 for some x ∈ X. ⇒ x = y-7 / 10 . Let us... 1
- Let f : W → W be defined as f(n) = n -1, if n is odd and f(n) = n + 1, if n is even. We are given, f : W → W, defined as \beginarrayll n-1, & n is odd \\ n+1, & n is even endarray. Let f(n1) =f(n2). If n1 is odd and n2 is eve... 1
- If f : R → R is defined by f(x) = x² – 3x + 2, find f[f(x)]. f(x) = x² – 3x + 2 f[f(x)] = [f(x)]² – 3[f(x)] + 2 = (x² – 3x + 2)² – 3(x² -3x + 2) + 2 = (x4 + 9x² + 4 – 6x³ – 12x + 4x²) – 3x² + 9x – 6 +... 1
- Show that the function f : R → { x ∈ R : – 1 – x < 1} defined by f(x) = \(\frac{x}{1+|x|}\) x ∈ R is one-one and onto. (i) Let x >0, f(x) = x / 1+x So, f(x1) = f(x2) ⇒ fracx₁1+x₁ = fracx₂1+x₂ or x1(1 + x2) = x2(1 + x1) or x1 + x1x2 = x2 + x1x2 ⇒ x1 = x2 When... 2
- Show that the function f : R → R given by f(x) = x³ is injective. f(x) = x³ ⇒ f(x1) = f(x2) ⇒ x³1 = x³1 ⇒ x1 = x2 ⇒ f is one-one, i.e., f is injective. 1
- Give examples of two functions f : N → Z and g : Z → Z such that gof is injective but g is not injective. Example: f(x) = x for f:mathbbN→mathbbZ , and g(x) = |x| for g:mathbbZ→mathbbZ . g is not injective: g(-1) = |-1| = 1 and g(1) = 1 , so two... 1
- Give examples of two functions f : N → N and g : N → N such that gof is onto but f is not onto. Example: f(x) = x + 1 and g(x) = begincases x - 1, & x > 1 \\ 1, & x = 1 endcases f is not onto: f(x) = x + 1 ≥ 2 for every x in mathbbN , s... 1
- Given a non-empty set X, consider P(X) which is the set of all such sets of X. Define the relation R in P(X) as follows: (i) A⊂A ⇒ R is reflexive. (ii) A ⊂ B, B ⊂ A ∴ R is not symmetric. (iii) If A ⊂ B, B ⊂ C, then A ⊂ C ∴ R is transitive. ⇒ R is not an equival... 1
- Given a non-empty set X, consider the binary operaion * : P(X) x P(X) → P(X), given by A * B = A ∩ B We have * : P(X) x P(X) → P(X) given by A* B = A ∩B ∴ X * A = X ∩ A = A * X = A for all A. ∴ X is an identity element. Let I is an other ide... 2
- Find the number of all the onto functions from the set {1, 2, 3, …,n } to itself. Let Y: 1 2 3 4 …. n X: 1 2 3 4 …. n One of the elements of set Y (say 1) may have any one of the pre-images 1, 2, 3, … n, i.e., in n ways. T... 1
- Let S = {a, b, c] and T = {1, 2, 3}. Find F-1 of the following functions F from S to T1, if it exists: (i) F = {(a, 3) S = {a, b, c} and T = {1, 2, 3} (i) F = [(a, 3), (b, 2), (c, 1)} i.e., F(a) = 3, F(b) = 2, F(c) = 1 ⇒ F-1(3) = a, F-1(2) = b, F-1(1) = c F-1... 1
- Consider the binary operations *: R x R → R and o: R x R → R defined as a * b = |a-b| and a o b = a for all a, b ∈R We have: a * b = | a – b | and a o b = a. (i) (a) a * b = | a – b | and b * a = | b – a | = | a – b | ⇒ Operation * is commutative. (b) (a *... 1
- Given a non-empty set X, let * : P(X) x P(X) → P(X) be defined as A * B = (A – B) ∪ (B – A) for all A, B ∈ P(X) Φ is the empty set. The operation * is defined as A * B = (A – B) ∪ (B-A) Putting B = Φ, we get A * Φ = (A – Φ) ∪ (Φ – A) = A ∪ Φ = A Φ * A... 1
- Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as a * b = (i) e is the identity element, if a * e = e * a = a. a * 0 = a + 0, 0 * a = 0 + a = a. ⇒ a * 0 = 0 * a = a. ∴ 0 is the identity for the oper... 1
- Let A = {1,2,3}. Then, the number of relations containing {1, 2} and {1, 3} which are reflexive, symmetric but not A) 1 There is only one relation containing (1, 2) and (1, 3), which is reflexive and symmetric but not transitive is {1, 1}. 2
- Let A = {1, 2, 3}. Then, number of equivalence relations containing (1, 2) is (A) 1 (B) 2 (C) 3 (D) 4 (B) 2 There are two equivalence relations containing (1, 2). 1
- Let f : R → R be the Signum Function defined as f(x) = f : R → R, the Signum Function is defined as f(x) = \beginarrayll 1, & x>0 \\ 0, & x=0 \\ -1, & x<0 endarray. 2
- Number of binary operations on the set (1, 2} having 1 as the identity element is (A) 10 (B) 16 (C) 20 (D) 8 Answer: Only 2 such binary operations exist, so none of the given options is mathematically correct. (The textbook key gives (B) 16, which i... 1
- Determine whether or not each of the definitions of * given below gives a binary operation. In the event that (i) a > b, a * b = a – b > 0 which belongs to Z+. But if a < b, a * b = a – b < 0 which does not belong to Z+. ⇒ given operation * is not a... 2
- For each binary operation * defined below, determine whether * is commutative or associative: (i) On Z, operation * is defined as (a) a * b = a – b ⇒ b * a = b – a But a – b + b – a ⇒ a * b # b * a (b) a – (b – c) * (a-b)-c Binary ope... 1
- Let * be the binary operation on N given by a * b = L.C.M. of a and b. (i) Find 5 * 7 and 20 * 16. (ii) Is * commutative Binary operation * defined as a * b = L.C.M. of a and b. (i) 5 * 7 = L.C.M. of 5 and 7 = 35. 20 * 16 = L.C.M. of 20 and 16 = 80. (ii) a * b... 1
- Is * defined on the set {1,2,3,4,5} by a * b = L.C.M. of a and b as a binary operation. Justify your answer. The given set = {1, 2, 3, 4, 5}. Binary operation * is defined as a * b = L.C.M. of a and b. 4 * 5 = 20 which does not belong to the given s... 1
- Let * be the binary operation of N defined by a * b = H.C.F. of a and b. Is * commutative? Is * associative? Binary operation on set N is defined as a * b = H.C.F. of a and b. (a) We know that H.C.F. of a, b = H.C.F. of b, a. ∴ a * b = b * a Binary... 1