Discrete Math (see the attachment)

Discrete Math Test #4 Study Guide Name: ___________________

Show all step. Explain your logic. Use the appropriate mathematical language.

1. Define a relation T from R to R as follows: for all (x, y) ∈ R × R, x T y ⇔ y < x + 2

a) Is (2, 0) ∈ T ? Is (0, 3) ∈ T ? Is (−6, 1) ∈ T ? Is (−4, −5) ∈ T ? b) (b) Sketch the graph of T in the Cartesian plane.

2. Let A = {3, 4, 5, 6, 7} and define a relation R on A as follows: for all x, y ∈ A, x R y

⇔ 2 | (x − y). (a) Is 6 R 3? Is 4 R 6?

(b) Draw the directed graph of R.

3. Define a relation S on the set of positive integers as follows: for all positive integers

m and n, m S n ⇔ m | n. (a) Is S reflexive? Justify your answer.

(b) Is S symmetric? Justify your answer.

4. P 476 #27 1)

5. P475#15

6. In a certain state, license plates each consist of 2 letters followed by either 3 or 4

digits. How many different license plates are there that have no repeated letters or digits?

7. p581 #6

8. p537 #13

9. Let A, B, and C be events in a sample space S such that S = A ∪ B ∪ C. Suppose that P (A) = 0.4, P (B) = 0.5, and P (A ∩ B) = 0.3. Find each of the following.

(a) P (A ∪ B) (b) P (C)

c) P ( c

A ∪ c

B )

10. . If a graph has vertices of degrees 1, 1, 3, and 3, how many edges does it have?

Why?

11. P 640 #18, 21.

12. p658 # 12, p659 #28