InterviewSolution
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InterviewSolution
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Let `A={1, 2, 3,..., 9}`and `R`be the relation on `AxxA`defined by `(a , b)R (c , d)`if `a+d=b+c`for all `(a , b), (c , d) in AxxA`. Prove that `R`is an equivalence relation and also obtain theequivalence class `[(2, 5)]`. |
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Answer» (i) eflexivity `Let (a,b) in A xxA, then ` ` (a,b)in A xxAimplies a,b in A ` `implies a+b=b+a` `implies (a,b)R(a,b).` `therefore ` R is reflexive . (ii) symmetry Let `(a,b)R(c,d). then ` `(a,b)R(c,d)implies a+d=d+a` `implies c+b=d+a` `implies (c,d)R(a,b).` ` therefore ` R is symmetric . (iii) Transitivity Let `(a,b) R(c,d) and (c,d) R (e f) . then ` `(a,b) R(c,d)and (c,d) R(e,f)` `implies a+d =b+c and c+f=d+e` `implies a+d+c+f=b+c+d+e` `a+f=b+e` `implies (a,b)R(e.f).` `therefore ` R is transitive ., thus ,R is reflexive , symmetric and transitive . hence ,R is an equivalence relation . `[(2,5)={(a,b):(2,5)R(a,b)}` `={(a,b):2+b=5+a}={(a,b):b-a=3}.` `={(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)}.` |
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