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Page No 5: Question 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 ): 3 x  −  y  = 0} (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 as integer} (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 ...

1. Use Euclid’s division algorithm to find the HCF of: (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 225 Solutions: (iii) 867 and 225 As we know, 867 is greater than 225. Let us apply now Euclid’s division algorithm on 867, to get, 867 = 225 × 3 + 102 Remainder 102 ≠ 0, therefore taking 225 as divisor and applying the division lemma method, we get, 225 = 102 × 2 + 51 Again, 51 ≠ 0. Now 102 is the new divisor, so repeating the same step we get, 102 = 51 × 2 + 0 The remainder is now zero, so our procedure stops here. Since, in the last step, the divisor is 51, therefore, HCF (867,225) = HCF(225,102) = HCF(102,51) = 51. Hence, the HCF of 867 and 225 is 51. 2. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer. Sol:  Let a be any positive integer and b = 6. Then, by Euclid’s algorithm, a = 6q + r, for some integer q ≥ 0, and r = 0, 1, 2, 3, 4, 5, because 0≤r<6. Now substituting the value of r...