Question

Let S be the set of people in the United States. Test the following binary relations on S for reflexivity, symmetry, antisymmetry, and transitivity.

a. x rho y ---> x is the husband of y.
b. x rho y ---> x is the spouse of y.
c. x rho y ---> x has the same parents as y.
d. x rho y ---> x is the brother of y.

Answer 1

Explanation:

a. x rho y ---> x is the husband of y.

Not reflexive, because x cannot be husband of x, not symmetric also not transitive

b. x rho y ---> x is the spouse of y.

Not reflexive, but symmetric because when x is the spouse of y, y is the spouse of x. Canot be transitive

c. x rho y ---> x has the same parents as y.

This is reflexive, symmetric and also transitive. Because when x and y have same parents and y,z have same parents then x and z have same parents.

d. x rho y ---> x is the brother of y

Not reflexive, need not be symmetric if x is a girl.

Transitive is true.