Question

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 1

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.