let A=(1,2,3,.....,9) and R be the relation in AxA defined by (a,b)R(c,d) if a+b=b+c for (a,b),(c,d) inAxA. Prove that R is an equivalence relation. Also obtain the equivalance class of ((2,5)). Share with your friends Share 12 Akhil Goyal answered this To prove equivalence, we need to show that the given relation is (i) Reflexive, (ii) Symmetric and (iii) Transitive.(i) For reflexivity, we need to show:∀(a,b)∈A×A, (a,b)R(a,b)Indeed this is true since, a+b=a+b ∀(a,b)∈A×A(ii) For Symmetric, we need to show: ∀(a,b), (c,d)∈A×A, (a,b)R(c,d) ⇒(c,d)R(a,b)Indeed this is also true as, (a,b)R(c,d) ⇒a+b=c+d ⇒c+d=a+b ⇒(c,d)R(a,b)(iii) For Transitive, we need to show: ∀(a,b), (c,d), (e,f)∈A×A, (a,b)R(c,d) & (c,d)R(e,f) ⇒(a,b)R(e,f)This is also true as, (a,b)R(c,d) ⇒a+b=c+dand, (c,d)R(e,f) ⇒c+d=e+fthus, a+b=c+d=e+f, ie, a+b=e+f⇒(a,b)R(e,f)Hence, relation R is an equivalence relation on A×ALet B⊂A×A be equivalent class of (2,5):B:=(x,y) ∈A×A | (x,y)R(2,5)or, B:=(x,y) ∈A×A | x+y=7substiting different values of x from A, we get:B={(1,6), (2,5), (3,4), (4,3), (5,2), (1,6)} 30 View Full Answer