First slide

Relations XII

Question

Let $W$ denote the words in the English
Dictionary. Define the relation $R b y R = {(x, y) \in W x W :$
the words $x$ and $y$ have at least one letter common}, then $R$ is

Moderate

A

reflexive, not symmetric and transitive
B

not reflexive, symmetric and transitive
C

reflexive, symmetric and not transitive
D

reflexive, symmetric and transitive

Solution

$(x, x) \in R V x \in W$ as all letters in both
are common. If $(x, y) \in R$ then $x$ and $y$ have a letter in common
$\Rightarrow$ $(y, x) \in R .$
Next, let $x = f i x, y = s i x$ and $z = s o n t h e n (x, y) \in R,$
$(y, z) \in R b u t (x, z) \notin R$
So $R$ is reflexive, symmetric but not transitive

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