2. Define a relation R on pairs of real numbers as follows: (a,b)R(c, d) iff either a < c or both a = c and b ? d. Is R apartial order? Why or why not? If R is a partial order, draw adiagram of some of its elements.

3. Define a relation R on integers as follows: mRn iff m + n iseven. Is R a partial order? Why or why not? If R is a partialorder, draw a diagram of some of its elements.

4. Define a relation R as follows:

R = {(a, a),(b, b),(c, c),(d, d),(c, a),(a, d),(c, d),(b, c),(b,d),(b, a)}

Is R a partial order? Why or why not? If R is a partial order,draw a diagram of some of its elements.

5. How many different partial ordering relations are there onthe set {a, b, c}?

6. Which of the following relations are partial orderings? Whichare total orderings? Which are well-orderings?

           a. The relation described in Problem 2

           b. The relation described in Problem 3

           c. The relation described in Problem 4

           d. The "less-than" relation on the integers

           e. The "less-than-or-equal" relation on the integers

           f. The "less-than-or-equal" relation on the natural numbers

           g. The "less-than-or-equal" relation on the real numbers

           h. The "less-than-or-equal" relation on the non-negative realnumbers

           i. The relations in Problem 5