Let W denote the words in the English dictionary. Define the relation R by: R = \(x,y) in W x W mid the words x and y have at least one letter in common. Then R is

Let W denote the words in the English dictionary. Defin... | Relation (Mathematics)

Let

W

denote the words in the English dictionary. Define the relation

R

by:

R = \{(x,y) \in W \times W \mid

the words

x

and

y

have at least one letter in common}. Then

R

not reflexive, symmetric and transitive

reflexive, symmetric and not transitive

reflexive, symmetric and transitive

reflexive, not symmetric and transitive.

Correct Answer:

Explanation

(b) : Given relation $R$ such that $\\R = \{(x,y) \in W \times W \mid$ the word $x$ and $y$ have atleast one letter in common} $\\$ where $W$ denotes set of words in English dictionary $\\$ Clearly $(x,x) \in R \forall x in W$ $\\ \therefore(x,x)$ has every letter common $\\ \therefore R$ is reflexive $\\$ Let $(x,y) \in R$ then $(y,x) \in R$ as $x$ and $y$ have atleast one letter in common. $\\ \Rightarrow R$ is symmetric. $\\$ But $R$ is not transitive $\\ \therefore$ Let $x =$ DON, $y =$ NEST, $z =$ SHE then $(x,y) \in R$ and $(y,z) \in R.\\$ But $(x,z) \notin R$ . $\\ \therefore R$ is reflexive, symmetric but not transitive.

Explanation

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