Let w denote the words in the english dictionary. Define the relation R by: R = `{(x,y) in W xx W` | words x and y have at least one letter in common}. Then R is: (1) reflexive, symmetric and not transitive (2) reflexive, symmetric and transitive (3) reflexive, not symmetric and transitive (4) not reflexive, symmetric and transitive

Let w denote the words in the english dictionary. Define the relation

1.	Let w denote the words in the english dictionary. Define the relation R by: R = `{(x,y) in W xx W` \| words x and y have at least one letter in common}. Then R is: (1) reflexive, symmetric and not transitive (2) reflexive, symmetric and transitive (3) reflexive, not symmetric and transitive (4) not reflexive, symmetric and transitive
Answer» Given relation `R` such that `R = {(x, y) in W xx W \|` the word x and y have at least one letter in common`}`, where W denotes set of words in English dictionary. Clearly `(x ,x ) in R` for all `x in W`. `:.` `(x, x)` has every letter common, therefore `R` is reflexive. Let `(x, y) in R` then `(y, x) in R` as `y` and `x` have at least one letter in common, this implies, `R` is symmetric. Let `(x,y) in R` and `(y,z) in R`, then it is not necessary that both `x` and `z` have at least one letter in common. Let `x = ABC, y = CDE, z = EFG` As we can see from this example `x` and `z` have no common letters. `:. (x,z) !in R`. So, `R` is not transitive. `:.` option `(1)` is correct.