Chapter 1 Relations and Functions
- Let f: R → R be defined as f(x) = 3x. Choose the correct answer: (a) f is one-one onto. (b) f is many-one onto. (a) f is one-one onto. f : R → R is defined by f(x) = 3x. (a) f(x1) = 3x1, f(x2) = 3x2 f(x1) = f(x2) ⇒ f is one-one. (b) For every member y... 1
- Determine whether each of the following relations are reflexive, symmetric and transitive: (i) Relation R in the set A = {1, 2, ….. , 14) defined as R = {(x, y) : 3x – y = 0} (a) Put y = x, 3x – x ≠ 0 ⇒ R is not reflexive. (b) If 3... 2
- Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a ≤ b2} (i) R is not reflexive ∵ a is not less than or equal to a² for all a ∈ R, e.g., 1/2 is not less than 1/4 . (ii) R is not symmetric, since if... 1
- Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b): b = a + 1} (i) R is not reflexive : a ≠ a + 1. (ii) R is not symmetric : If b = a + 1, then a ≠ b + 1. (iii) R is not transitive :If b = a + 1, c = b +... 1
- Show that the relation R in R defined as R = {(a, b): a ≤ b} is reflexive and transitive but not symmetric. R = {(a, b): a ≤ b} (i) R is reflexive. Replacing b by a, a ≤ a ⇒ a = a is true. (ii) R is not symmetric : a < b and b < a which is not true... 1
- Check whether the relation R, defined by R = {(a, b) : a ≤ b3} is reflexive, symmetric or transitive. (i) a ≤ a³ is not true. e.g. 1/2 is not less than ( 1/2 )³ ∴ R is not reflexive. (ii) If a ≤ b³, then b is not less than or equal a3, e.g. 1... 1
- Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} (i) (1, 1), (2, 2), (3, 3) do not belong to relation R. ∴ R is not reflexive. (ii) It is symmetric because (1, 2) and (2, 1) belong to R. (i... 2
- Show that the relation R in the set A of all books in a library of a college, given by R = {(x, y): x (i) The number of pages in a book remain the same. ⇒ Relation R is reflexive. (ii) The book x has the same number of pages as the book y. ⇒... 1
- Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b): | a – b | is even} A = {1, 2, 3, 4, 5} and R = {(a, b): | a – b | is even} R = {(1, 3), (1, 5), (3, 5), (2, 4), (3, 1), (5, 1), (5, 3), (4, 2)} (a) (i) Let us... 1
- Show that the relation R in the set A = [x ∈ Z : 0 ≤ x ≤ 12}, given by (i) R = {(a, b): | a – b | is a multiple of 4} The set A = {x ∈ Z : 0 ≤ x ≤ 12} = {0, 1, 2, ………, 12} (i) R = {(a, b): | a – b | is a multiple of 4} |a – b | = 4k or b = a + 4k. ∴ R = {(1,... 2
- Give examples of relations which are (i) Symmetric but neither reflexive nor transitive. (ii) Transitive but neither Let A = set of straight lines in a plane. (i) R : {(a, b): a is perpendicular to b] Let a and b be two perpendicular lines. (a) If line a is... 2
- Show that the relation R in the set A of points in a plane, given by R: {(P, Q): Let O be the origin, then the relation R = {(P, Q): OP = OQ} (i) R is reflexive. Take any distance OP. We know OP = OP. ∴ R is reflexive. (i... 1
- Show that the relation R, defined in the set A of all triangles as R = {(T1, T2) : T1 is similar to T2} (i) In a set of triangles, R = {(T1, T2): T1 is similar to T2}, (a) A triangle T1 is similar to itself. R is reflexive. (b) If triangle T1 i... 1
- Show that the relation R, defined in the set A of all polygons as R = {(P1, P2) : P1 and P2 Let n be the number of sides of a polygon P1. R = {(P1, P2): P1 and P2 are n sided polygons}. (i) (a) Any polygon P1 has n sides ⇒ R is refl... 1
- Let L be the set of all the lines in XY-plane and R be the relation in L defined as R = {(L1, L2) L = Set of all the lines in XY-plane. R = {(L1, L2): L1 is parallel to L2} (i) (a) L1 is parallel to itself ⇒ R is reflexive. (b) L1 is para... 1
- Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1,1), (4,4), (1,3), (3, 3), (3, 2)} (b) R is reflexive and transitive but not symmetric. Explanation:: Let R be the relation with set {1,2, 3,4}, given by R = {(1,2), (2,2), (1... 1
- Let R be the relation in the set N given by R = {(a, b): a = b – 2, b > 6}. (C) (6, 8) ∈ R Explanation: According to given condition b>6 {.: option a cannot be answer and a is smaller than b therefore option d is als... 1