A+ Answers

 Discrete Review

1. A relation R over the set S = {x, y, z} is defined by:

  {(x, x), (x, y), (y, x), (x, z), (y, z), (y, y), (z, z)}

 What properties hold for this relation? Please provide only one answer

 A. Symmetry only

 B. Reflexivity only

 C. Reflexivity and Symmetry

 D. Antisymmetric

 E. Reflexivity and Transitivity

2. Let A and B be any sets. Prove the following set identity using the laws of set

 theory (set identities). Justify each step with the law that you used.

3. Let the relation R = {(0, 0), (0, 3), (1, 0), (1, 2), (2, 0), (3, 2)}

 Find R’, the transitive closure of R.

4. Using the predicate symbols shown and appropriate quantifiers, write each

 English language statement as a predicate wff. (The domain is the whole world.)

  B(x) is “x is a ball”

  R(x) is “x is round”

  S(x) is “ x is a soccer ball”

 A) All balls are round.

B) Not all balls are soccer balls.

 C) All soccer balls are round.

 D) Some balls are not round.

5. For all integers n = 1, prove the following statement using mathematical

 induction.

 A) Prove the base step:

B) State the inductive hypothesis:

C) State what you have to show:

D) Proof:

6. Prove the expression below is a valid argument using the deduction method (that

 is, using equivalences and rules of inference in a proof sequence).

7. Let S {0, 2, 4, 6, 8} and T = {1, 3, 5, 7}.

 Determine whether each of the following sets of ordered pairs is a function with

 domain S and co-domain T.

A) {(2, 1), (4, 5), (6, 3)}

 B) {(0, 2), (2, 4), (4, 6), (6, 0), (8, 2)}

C) {(6, 3), (2, 1), (0, 3), (8, 7), (4, 5)}

D) {(2, 3), (4, 7), (0, 1), (6, 5), (8, 7)}

E) {(6, 1), (0, 3), (4, 1), (0, 7), (2, 5), (8, 5)}

8. How many numbers must be selected from the set {1, 3, 5, 7, 9, 11, 13, 15} to

 guarantee that at least one pair of these numbers add up to 16? Explain.

9. A set of 4 coins is selected from a box containing 8 dimes and 6 quarters. Show

 all work.

 A) Find the number of sets of four coins:

B) Find the number of sets in which two are dimes and two are quarters:

 C) Find the number of sets composed of all dimes or all quarters:

D) Find the number of sets with three or more quarters:

10. For each of the following characteristics, determine whether the graph exists or

 why such a graph does not exist.

 A) Simple graph with seven nodes, each of degree 3.

B) Four nodes, two of degree 2 and two of degree 3.

C) Three nodes of degree 0, 1, and 3, respectively.

D) Complete graph with four nodes, each of degree 2.

11. Draw the minimum-weight spanning tree (or give a list of edges) for the graph

 below using Kruskal’s algorithm.

Calculus Review

1. Find the limit if it exists.