Question

Let W denote the words in the English dictionary. Define the relation R by R=(x,y)ϵW×W Then R is

A. not reflexive, symmetric and transitive
B. relexive, symmetric and not transitive
C. relexive, symmetric and transitive
D. reflexive, not symmetric and transitive

Answer 1

The relation R defined by R=(x,y)ϵW×W is reflexive, symmetric, and transitive.

Step-by-step explanation:

The relation R defined by R = the words x and y have at least one letter in common is reflexive, symmetric, and transitive. So the correct answer is option C.

To show that R is reflexive, we need to prove that (x, x) ∈ R for all x ∈ W. Every word has at least one letter in common with itself, so (x, x) ∈ R, making R reflexive.

To show that R is symmetric, we need to prove that if (x, y) ∈ R, then (y, x) ∈ R for all x, y ∈ W. If x and y have at least one letter in common, then y and x will also have at least one letter in common, so (y, x) ∈ R, making R symmetric.

To show that R is transitive, we need to prove that if (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R for all x, y, z ∈ W. If x and y have at least one letter in common, and y and z have at least one letter in common, then x and z will also have at least one letter in common, so (x, z) ∈ R, making R transitive.