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1 votes
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 A relation $R$ is defined on $N \times N$, such that $(a,b) R (c,d)$ iff $a+d = b+c.$ The relation $R$ is

  1. reflexive but not transitive            
  2. reflexive and transitive but not symmetric                 
  3. an equivalence relation                  
  4. a partial order

1 Answer

4 votes
4 votes
$(a,b) R (c,d)$ iff $a+d = b+c$.

So,

$(c,d) R (a,b)$ iff $c+b = d+a$.

From property of addition, $d+a = a + d, c+b = b+c$.

So, $(a,b) R (c,d)$ iff $(c,d) R (a,b)$. Hence, $R$ is symmetric.

$a+b = b+a$, so $(a,b) R (a,b)$. Hence, $R$ is reflexive.

If, $(a,b) R (c,d)$ and $(c,d) R (e,f)$, we have $a+d = b+c$ and $c+f = d+e$. So, $a + d + c + f = b + c + d + e \implies a + f = b + e \implies (a,b) R (e,f).$

Hence, $R$ is transitive.

Since $R$ is reflexive, transitive and symmetric, it is an equivalence relation.
Answer:

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