# IST 230 IST230 Final Exam with Answers

IST 230 IST/230 IST230 FINAL EXAM

**Chapter 1**

1.

a. State in English the converse of “If it is raining, then my lawn is wet.” **(5 points)**

b. State in English the contrapositive of “If it is raining, then my lawn is wet.” **(5 points)**

c. Let the domain of discourse be the set of people in our IST 230 class. Define the following predicate:

S(x): x is less than five feet tall.

Translate the following English statement into a logical expression with the same meaning.**(5 points)**

“It is not the case that someone in our class is at least five feet tall.”

**Chapter 2**

2. **True or False**: If A = {1, 2, 3} and B = {Turtle, Wax}, then (1, 2) is an element of AxB (where x indicates the cross product). **(5 points)**

**Chapter 3**

3a. Let the function f from the *positive integers* to the *positive* *integers* be defined by f(x) = x*x (where * is ordinary integer multiplication). Explain why f is or is not onto. **(5 points)**

3b. Let sets A and B be defined as follows: A = {a1} and B = {b1, b2}. List as set(s) of ordered pairs *all* the functions from A to B (each such function should be listed as one set of ordered pairs), and explain whether any function from A to B is onto. **(5 points)**

3c. Let A = {a1, a2} and B = {b1, b2, b3}. Let the function f:A → B be given by the following set of ordered pairs: f = {(a1,b2),(a2,b3)}. **(10 points)**

List as a set of ordered pairs a function g: B→A with the property that for all a in A g(f(a)) = a, and show that this property holds. Note that since the domain of g is B, you need to make sure your function g maps **each**element in B to some element in A.

**Chapter 5**

**5. **Let the set A be defined as A = {a, b, c, d}, and let the relation R on the set A be defined by

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

Explain why R is or is not symmetric, and why R is or is not reflexive. (Don't confuse the two!) **(5 points)**

**Chapter 6**

6. What is the value of the variable count after all the loops in the following pseudocode execute? **(5 points)**

count:=0

For i= 1 to 2

For j=1 to 2

count:=2j(count-1)

End-for

End-for

count = 0 to start

**Chapter 7**

7. Find the next three terms (terms a_{2}, a_{3}, a_{4}) of the sequence defined as follows

a_{1} = 0

a_{k} = 3 + 2a_{k-1} for k >= 2**(5 points)**

**Chapter 8**

8a. Show calculations and determine how many digits are needed to represent the base 5 expansion of 4096 (where 4096 is in base 10). Then write the base 5 expansion of 4096. **(5 points)**

8b. Explain why a multiplicative inverse of 7 mod 13 does or does not exist. *If one does exist, calculate it, and explain why there is or is not only one such multiplicative inverse*. If one does not exist, explain why not.**(5 points)**

**Chapter 9**

9. A husband and wife and their two children line up for a photo. How many ways are there for these four people to line up so that the husband and wife are next to each other? **(10 points)**

**Chapter 11**

11. An experiment is performed to flip a fair coin 10 times and observe the outcome of each flip: heads (labeled 'H') or tails (labeled 'T'). For instance, one outcome, written as a 10-tuple, might be (H,T,T,T,T,T,T,H,H,H). What is the probability of obtaining an outcome that has exactly one heads? **(10 points)**

**Chapter 12**

12. Let vertex sets V1 and V2 be defined by V1= {1, 2, 3} and V2 = {a, b, c}. Let E1 = { { 1, 2}, {2, 3} }, and let E2 = { {a, b}, {b, c} } be the edge sets corresponding to the vertex sets V1 and V2, respectively. Write as a set of ordered pairs a function f:V1→V2 that is a *bijection* from V1 to V2. **(10 points)**

**Chapter 13**

13. Let V = {a, b, c, d, e} be a vertex set and E = { {a,b}, {b,c}, {c,d}, {d,e}, {e,c}} be the edge set corresponding to V. **(5 points)**

True or False: The pair (V, E) is a tree. (Hint: draw it and see what it looks like.)