Prove that the collection of all intervals in R with endpoints in QU{to} is countable.

16. Let and define on by if and only if . Determine whether is reflexive, symmetric, or transitive.

21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in Exercise 2 are irreflexive? 2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric, or transitive. Justify your answers. a. if and only if b. if and only if c. if and only if for some in . d. if and only if e. if and only if f. if and only if g. if and only if h. if and only if i. if and only if j. if and only if. k. if and only if.

Show that the converse of Eisenstein’s Irreducibility Criterion is not true by finding an irreducible such that there is no that satisfies the hypothesis of Eisenstein’s Irreducibility Criterion.