# 1. Determine whether each of the following relations are reflexive, symmetric and transitive:(i) Relation $R$ in the set $A = \{1,2,3 ...,13 ,14\}$ defined as$R = \{(x,y): 3x - y = 0\}$

$A = \{1,2,3 ...,13 ,14\}$

$R = \{(x,y): 3x - y = 0\}$ $= \left \{ \left ( 1,3 \right ),\left ( 2,6 \right ),\left ( 3,9 \right ),\left ( 4,12 \right ) \right \}$

Since,  $\left ( 1,1 \right ),\left ( 2,2 \right ),\left ( 3,3 \right ),\left ( 4,4 \right ),\left ( 5,5 \right )\cdot \cdot \cdot \cdot \cdot \cdot \left ( 14,14 \right ) \notin R$ so $R$ is not reflexive.

Since, $\left ( 1,3 \right ) \in R$ but  $\left ( 3,1 \right ) \notin R$ so $R$ is not  symmetric.

Since, $\left ( 1,3 \right ),\left ( 3,9 \right ) \in R$ but $\left ( 1,9 \right ) \notin R$ so $R$ is not  transitive.

Hence, $R$ is neither reflexive nor  symmetric and nor transitive.

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