Question

Let 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 transitive relation

Answer 1

Hey mate....

here's the answer....

Let (a,b)R(c,d)and(c,d)R(e,f

)(a,b)R(c,d)and(c,d)R(e,f)

a+b=b+ca+b=b+c

and c+f=d+ec+f=d+e

a+b=b+ca+b=b+c

=>a−c=b−d(1)c+f=d+e(2)

=>a−c=b−d(1)c+f=d+e(2)

adding (1) and (2)

a−c+c+f=b−d+d+ea

−c+c+f=b−d+d+e

a+f=b+ea+f=b+e

=>(a,b)R(e,f)

=>(a,b)R(e,f)

R is transitive

Hope this helps❤

Answer 2

$\huge\mathbb\pink{Answer}$

For all (a,b) (c,d) (e,f) belong to A×A

we have

(a,b)R(c,d) and (c,d) R( e,f)

a+d= b+c and c+f= d+e

a+d+c+f= b+c+d+e

a+f =b+e

(a,b) R(e,f)

Thus R is transitive on A×A