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Question

# If A={1,2,3,....,9} and R be the relation in A×A defined by (a,b)R(c,d). If a+d=b+c for (a,b),(c,d) in A×A. Prove that R is an equivalence relation. Also, obtain the equivalence class [(2,5)].

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Solution

## (i) Reflexive : (a,b)R(c,d) if (a,b)(a,d)∈A×Aa+b=b+cConsider (a,b)R(a,b)⇒(a,b)∈A×A⇒a+b=b+a Hence R is reflexive(ii) Symmetric : (a,b)R(c,d) given by (a,b)(c,d)∈A×Aa+d=b+c⇒c+b=d+a⇒(c,d)R(a,b) Hence R is symmetric(iii) equivalence relation = reflexive + symmetric + transitiveTransitive : Let (a,b)R(c,d) and (c,d)R(c,f)(a,b)(c,d),(c,f)∈A×Aa+b=b+c and 4c+f=d+ea+b=b+c⇒a−c=b−d...(1) & c+f=d+e...(2)adding (1) & (2) a+f=b+e⇒(a,b)R(e,f) Transitive So, R is equivalence relation from A={1,2,3...9} are will select a and b such that 2+b=5+aLet b=4 and a=1 Hence (2+4)=(5+1)and [(2,5)]={(1,4)(2,5),(3,6)(4,7)(5,8)(6,9)} is the equivalent class under relation R

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