Let W denote the words in the English dictionary- Define the relation R by R - x - y - W - W - the words x and y have at least one letter in common- Then R isA- not reflexive- symmetric and transitiveB- relexive- symmetric and not transitiveC- relexive- symmetric and transitiveD- reflexive- not symmetric and transitive

The correct option is A not reflexive, symmetric and transitiveClearly (x,x) ϵ R ∀ x ϵ W. So R is relexive. Let (x,y) ϵ R then (y,x) ϵ R as x and y have at l ...

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Standard XII

Mathematics

Equivalence Relation

Let W denote ...

Question

Let W denote the words in the English dictionary. Define the relation R by $R = {(x, y) ϵ W \times W |$ the words x and y have at least one letter in common.} Then R is

not reflexive, symmetric and transitive

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relexive, symmetric and not transitive

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relexive, symmetric and transitive

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reflexive, not symmetric and transitive

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Solution

The correct option is A not reflexive, symmetric and transitive
Clearly $(x, x) ϵ R \forall x ϵ W$ . So R is relexive.
Let $(x, y) ϵ R t h e n (y, x) ϵ R$ as x and y have at least one letter in common. So, R is symmetric.
But R is not transitive for example
Let x = INDIA, y = BOMBAY and z = JOKER then (x, y) $ϵ$ R (A is common) and (y, z) $ϵ$ R (O is common) but (x, z) $\notin$ R. (as no letter is common)

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Let W denote the words in the English dictionary. Define the relation R by R={(x,y)ϵW×W| the words x and y have at least one letter in common.} Then R is

Let W denote the words in the English dictionary. Define the relation R by $R = {(x, y) ϵ W \times W |$ the words x and y have at least one letter in common.} Then R is