# Let $\dpi{100} W$ denote the words in the English dictionary. Define the relation $\dpi{100} R$  by :$\dpi{100} R=\left \{ \left ( x,y \right )\epsilon W\times W\mid the\, word\, x\, and\, y\, have\, at\, least\, one\, letter\, in\, common\, \right \}\; Then \; R\; is$ Option 1) not reflexive, symmetric and transitive Option 2) reflexive, symmetric and not transitive Option 3) reflexive, symmetric and transitive Option 4) reflexive, not symmetric and transitive.

As we learnt in

REFLEXIVE RELATION -

A relation R in A is said to be reflexive,  if a R a ,∀ a ∈ A

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SYMMETRIC RELATION -

A relation R in A is said to be symmetric, if a R b ⇒ b R a,∀ a,b ∈ A

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TRANSITIVE RELATION -

A relation R in A is said to be transitive, if a R b and b R c ⇒ a R c ∀ a,b,c ∈ A

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$R = [(x, y)\epsilon \omega \times \omega]$

x, y having at least one letter in common.

(x, x) both are same so it is reflexive.

$(x,y), (y,x)\epsilon R$ so it is symmetric.

$(x,z), (z,y)\epsilon R$ so it is not transitive.

Correct option is 2.

Option 1)

not reflexive, symmetric and transitive

This is an incorrect option.

Option 2)

reflexive, symmetric and not transitive

This is the correct option.

Option 3)

reflexive, symmetric and transitive

This is an incorrect option.

Option 4)

reflexive, not symmetric and transitive.

This is an incorrect option.

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