First slide

Relations XII

Question

Let R be a relation over the set N x N and it is defined by $(a, b) R (c, d) \Rightarrow a + d = b + c$ Then, R is

Moderate

A

Reflexive
B

Symmetric
C

Equivalence
D

Transitive

Solution

We have, (a, b) R (a, b)for all $(a, b) \in N \times N \Rightarrow R$ is reflexive
R is symmetric for we have $(a, b) R (c, d) \Rightarrow a + d = b + c$
$\begin{array}{ll} \Rightarrow & d + a = c + b \\ \Rightarrow & c + b = d + a \Rightarrow (c, d) R (a, b) \end{array}$
Hence, R is symmetric.
Then, by definition of R, we have
a+ d = b + c and c+ f= d + e
hence by addition, we get
a+d+ c+ f =b+ c + d+e
or a+f =b +e
Hence, (a b) R (e, f )
Thus, (a, b) R (c, d) and (c, d)R (e, f)
$\Rightarrow$ (a, b)R (e f)
Hence, R is transitive.
Clearly R is equivalence

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