NCERT Class 12 Maths · Chapter 1 Relations And Functions · Exercise 1.1 Question 1

✅ Step-by-Step Solution

1. Use: reflexive means every (a,a) belongs; symmetric means aRb⇒bRa; transitive means aRb and bRc⇒aRc. One counterexample disproves a property.
2. (i) R={(1,3),(2,6),(3,9),(4,12)}. (1,1) is absent, (1,3) has no reverse, and (1,3),(3,9) belong while (1,9) does not. Thus none of the three properties holds.
3. (ii) R={(1,6),(2,7),(3,8)}. It has no diagonal pairs and no reverse of (1,6). No second coordinate 6,7,8 is a first coordinate of another pair, so no two-pair chain exists. The implication defining transitivity is vacuously true.
4. (iii) Every positive integer divides itself. If a divides b and b divides c, then a divides c. Symmetry fails since 1 divides 2 but 2 does not divide 1. Therefore reflexive and transitive only.
5. (iv) The difference of every two integers is an integer, so R=ℤ×ℤ, which is reflexive, symmetric and transitive.
6. (v)(a),(b) Under the stated single-assignment model, equality of workplace or locality is reflexive, symmetric and transitive. Multiple or missing workplace assignments would require a different model.
7. (v)(c) Nobody is 7 cm taller than themselves; reversing a 7 cm difference is impossible; two successive 7 cm differences give 14 cm, not 7 cm. All three properties fail when such height comparisons are available.
8. (v)(d) Under the stated textbook role convention there are no diagonal or reversed wife pairs, and no wife-of-wife chain, so transitivity is vacuous. (e) Fatherhood has no diagonal or reverse, while father-of-father is grandfather, not father; all three fail in the usual family model.