 # IST 230 IST230 Final Exam with Answers

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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 eachelement 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 a2, a3, a4) of the sequence defined as follows

a1 = 0

ak = 3 + 2ak-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.)

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