Question

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

3. Define a relation R on integers as follows: mRn iff m + n is even. Is R a partial order? Why or why not? If R is a partial order, 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 on the set {a, b, c}?

6. Which of the following relations are partial orderings? Which are 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 real numbers

            i. The relations in Problem 5

Answer 1