Ncert Solutions for Class 12 Mathematics Chapter 1: Number System Exercise 1.1

Q1. Determine whether each of the following relations are reflexive, symmetric and transitive: (ii) Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x, y) : y is divisible by x} (iv) Relation R in the set Z of all integers defined as R = {(x, y) : x – y is an integer}   Q 1. Determine whether each of the following relations are reflexive, symmetric and transitive: (i) Relation R in the set A = {1, 2, 3, ..., 13, 14} defined as R = {(x, y) : 3x – y = 0}   Q 1.Determine whether each of the following relations are reflexive, symmetric and transitive: (v) Relation R in the set A of human beings in a town at a particular time given by (a) R = {(x, y) : x and y work at the same place} (b) R = {(x, y) : x and y live in the same locality} (c) R = {(x, y) : x is exactly 7 cm taller than y} (d) R = {(x, y) : x is wife of y} (e) R = {(x, y) : x is father of y}   2. Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a ? b2} is neither reflexive nor symmetric nor transitive.   3. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.   4. Show that the relation R in R defined as R = {(a, b) : a ? b}, is reflexive and transitive but not symmetric.   5. Check whether the relation R in R defined by R = {(a, b) : a ? b3} is reflexive, symmetric or transitive.   6. Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.   7. Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y) : x and y have same number of pages} is an equivalence relation.   8. Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : a – b is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.   9. Show that each of 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} (ii) R = {(a, b) : a = b}  
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