-2

Let $A$ be the set of integer ordered pairs.

Define relation $R$ on $A$ by $(a,b)R(c,d)$ $\iff$ $a+d = b+c$.

Prove that $R$ is an equivalence relation.

Community
  • 1
user3754317
  • 3
  • Please format your question and describe what you tried and where you are stuck. – avs Dec 07 '16 at 19:24
  • 1
    An equivalence relation has 3 properties (by definition). What are they? How would you test each. Does R have any or all of these properties? – Doug M Dec 07 '16 at 19:26

2 Answers2

0

If you can find a function $f$ such that $f(a,b) = f(c,d)$ iff $(a,b)R(c,d)$ then you can show that $R$ is an equivalence relation (because, in some sense, you have reduced to the problem to a known equivalence relation $=$).

The equivalence relation in the question suggests a fairly straightforward $f$.

copper.hat
  • 172,524
0
  • Reflexive

$a+b=b+a\implies (a,b)$R$(a,b)$

  • Symetric

$a+d=b+c\implies c+b=d+a$ $\implies (c,d)$R$(a,b)$

  • Transitive

$a+d=b+c$ and $c+f=d+e$ $ \implies a+d+c+f=b+c+d+e$

$\implies a+f=b+e\implies$ $(a,b)$R$(e,f)$.

hamam_Abdallah
  • 62,951