4. Prove that the next relation is equivalence relation on the set of all people: {(x, y)|x and y are the same age}.
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- 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.In Exercises 610, a relation R is defined on the set Z of all integers. In each case, prove that R is an equivalence relation. Find the distinct equivalence classes of R and list at least four members of each. xRy if and only if x+3y is a multiple of 4.Prove Theorem 1.40: If is an equivalence relation on the nonempty set , then the distinct equivalence classes of form a partition of .
- Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.In Exercises , prove the statements concerning the relation on the set of all integers. 17. If and , then .In Exercises , prove the statements concerning the relation on the set of all integers. 18. If and , then .