First slide

Relations XII

Question

On Z, the set of integers define a relation R as follows $a, b \in Z, aRb if 3 ∣ (2 a + b)$ then

Moderate

A

R is reflexive, symmetric but not transitive
B

R is reflexive, transitive but not symmetric
C

R is anti-symmetric
D

R is an equivalence relation

Solution

If $a \in Z, aRa as 3 ∣ (2 a + a)$
R is reflexive.
Suppose a R b
$\begin{aligned} \Rightarrow 3 ∣ (2 a + b) \\ \Rightarrow 3 ∣ (3 a + b - a) \\ \Rightarrow 3 | (b - a) \Rightarrow 3 | (a - b) \\ \Rightarrow 3 | (3 b + a - b) \Rightarrow 3 | (2 b + a) \\ \Rightarrow b ~ a . \\ \Rightarrow R is symmetric. \end{aligned}$
Next, suppose a R b, b R c
$\Rightarrow 3 | (2 a + b), 3 | (2 b + c)$
$\begin{array}{ll} \Rightarrow & 3 | (b - a), 3 | (c - b) \\ \Rightarrow & 3 ∣ (b - a + c - b) \\ \Rightarrow & 3 ∣ (c - a) \end{array}$
$\begin{array}{ll} \Rightarrow & 3 ∣ (c - a + 3 a) \\ \Rightarrow & 3 ∣ (2 a + c) \\ ∴ & aRc \end{array}$
R is transitive.
Hence, R is an equivalence relation

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