All questions
- 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
- Let * be a binary operation on the set Q of rational numbers as follows: (i) a * b = a – b Operation * is on the set Q. (i) Defined as a * b = a – b (a) Now b * a = b – a. But a – b ≠ b – a ∴ a * b ≠ b * a ∴ Operation * is not comm... 1
- Show that none of the operations given above has identity. The binary operation * on set Q is (i) defined as a * b = a – b. For idenity element e, a * e = e * a = a. But a * e = a – e # a and e * a =... 2
- Let A = N x N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c,b+d) A = N x N Binary operation * is defined as (a, b) * (c, d) = (a + c, b + d). (a) Now, (c, d) * (a, b) = (c + a, d + b) = (a + c, b + d) ⇒ (a... 1
- State whether the following statements are true or false. Justify. (i) For any arbitrary binary operation * on a set N (i) A binary operation on N is defined as a * a = a ∀a ∈ N. Here, operation * is not defined. ∴ Given statement is false. (ii) * is a binary... 2
- Consider a binary operation * on N defined as a * b = a³ +b³. Choose the correct answer: (B) Is * commutative but not associative The binary operation * on set N is defined as a * b = a³ +b³ Now, b * a = b³ + a³ = a³ +b³. ⇒ a * b... 1
- Let f : {1. 3 4} → {1,2, 5} and g: {1, 2, 5} → {1,3} be given by f = {(1, 2), (3, 5), (4,1)} f= {(1,2), (3,5), (4,1)} g = {(I, 3), (2, 3), (5, 1)} (f) = 2, g(2) = 3 ⇒ gof(1) = 3 f(3) = 5, g(5) = 1 ⇒ gof(3) = 1 f(4) = 1, g(1) = 3 ⇒ go... 1
- Find gof and fog, if \(x^{\frac{1}{3}}\) (i) f(x) = |x|, and g(x) = |5x – 2|. (i) f(x) = |x|, and g(x) = |5x – 2| gof(x) = g[f(x)]=g|x| = |5|x|-2|. fog(x) = f[g(x)] = f(|5x- 2|) =||5x-2|| = |5x – 2|. (ii) f(x) = 8x³ an... 1
- State with reasons whether each of the following functions has a inverse or not: (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,... 1
- Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f. f : R → R given by f(x) = 4x + 3. f(x1) = 4x1 + 3, f(x2) = 4x2 + 3 If f(x1) = f(x2), then 4x1 + 3 = 4x2 + 3 or 4x1 = 4x2 ⇒ x1 = x2. ⇒ f is o... 1
- Consider f : R+ → [4, ∝) given by f(x) = x² + 4. Show that f is invertible with the inverse of f-1 of f f(x1) = x²1 + 4 and f(x2) = x²2 + 4 So, f(x1) = f(x2) ⇒ x²1 + 4 = x²2 + 4 or x²1 = x²2 ⇒ x1 = x2 As x ∈ R, ∴ x > 0, x²1 = x²2 ⇒ x1 = x2 ⇒ f... 1
- Let f : X → Y be an invertible function. Show that f has a unique inverse. If f is invertible, then gof(x) = Ix and fog(y) = Iy. ⇒ f is one-one and onto. Let there be two inverses g1 and g2 ∴ fog1 (y)= Iy, f0g2(y) =... 1