4. (a) Let R be the set of all real numbers. Consider a relation S on R, i.e. SCRX R. which is defined as follows: S = {(r.y) rye R.ry E Z}. Prove that S is an equivalence relation.

Does the relation is in love with have a reflexive property consider one person? a symmetric property consider two people? a transitive property consider three people?

In each of the following parts, a relation is defined on the set of all human beings. Determine whether the relation is reflective, symmetric, or transitive. Justify your answers. xRy if and only if x lives within 400 miles of y. xRy if and only if x is the father of y. xRy if and only if x is a first cousin of y. xRy if and only if x and y were born in the same year. xRy if and only if x and y have the same mother. xRy if and only if x and y have the same hair colour.

Label each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.