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Question

# Let A={1,2,3,....9} and R be 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

## A={1,2,3...9}R in A×A(a,b) R (c,d) if (a,b)(c,d) ∈ A∈Aa+b=b+cConsider (a,b) R (a,b) (a,b)∈A×Aa+b=b+aHence, R is reflexive.Consider (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.Let (a,b) R (c,d) and (c,d) R (e,f)(a,b),(c,d),(e,f),∈A×Aa+b=b+c and c+f=d+ea+b=b+c⇒a−c=b−d-- (1)c+f=d+e-- (2)Adding (1) and (2)a−c+c+f=b−d+d+ea+f=b+e(a,b)R(e,f)R is transitive.R is an equivalence relation.We select from set A={1,2,3,....9}a and b such that 2+b=5+aso b=a+3Consider (1,4) (2,5) R (1,4)⇒2+4=5+1[(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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