First slide

Relations XII

Question

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

Moderate

A

Reflexive only
B

Symmetric only
C

Transitive only
D

An equivalence relation

Solution

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

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