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Answered: - I'm completely lost on this, i finished #1. It is due tonight, im


I'm completely lost on this, i finished #1. It is due tonight, im panicking. HELP !


Midterm (Show/Explain all Work)

 


 

IST 230

 


 

Chapter 1

 

1.

 


 

Complete the following truth table for

 


 

( p ? ( ?r )) ? q .

 


 

(4 points)

 


 

p

 


 

q

 


 

r

 


 

?r

 


 

p? ( ?r )

 


 

( p ? ( ?r )) ? q

 


 

T

 

T

 

T

 

T

 

F

 

F

 

F

 

F

 


 

T

 

T

 

F

 

F

 

T

 

T

 

F

 

F

 


 

T

 

F

 

T

 

F

 

T

 

F

 

T

 

F

 


 

F

 

T

 

F

 

T

 

F

 

T

 

F

 

T

 


 

F

 

T

 

F

 

T

 

F

 

F

 

F

 

F

 


 

T

 

T

 

F

 

F

 

T

 

T

 

T

 

F

 


 

Chapter 2

 

2. In this question, the domain of discourse is the set of people in our IST 230 class. Define the following

 

predicates:

 

T(x): x is more than six feet tall.

 

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

 

Translate the following English statements into a logical expression with the same meaning.

 


 

a.

 


 

The negation of ?Everyone in our IST 230 class is more than six feet tall.? Translate in two ways:

 

one using a universal quantifier and one using an existential quantifier. (4 points)

 


 

b. ?No one in our class is less than five feet tall.? (2 points)

 


 

3.

 


 

Let A and B be defined as follows:

 

A = {Tom, Bo}

 

B = {Java, Python, R}

 

a. Write the cross product AxB as a set of ordered pairs. (4 points)

 


 

b. Write the cross product BxA as a set of ordered pairs. (4 points)

 


 

Midterm (Show/Explain all Work)

 


 

IST 230

 


 

c. Write one ordered pair that is in the set AxB but is not in the set BxA. (2 points)

 


 

Chapter 3

 

4.

 


 

Let sets A and B be defined as follows:

 

A = {Java, Python, R}

 

B = {Tom, Bo}

 


 

a. List as a set of ordered pairs one 1-1 function from A to B, or explain why no such function exists.

 

(5 points)

 


 

b.

 


 

List as a set of ordered pairs one onto function from A to B, or explain why no such function

 

exists. (5 points)

 


 

c.

 


 

Let f:B ? A be given by this set of ordered pairs:

 

f = { (Tom, Java), (Bo, Python) }

 

If f has an inverse function, call it g and list g as a set of ordered pairs. If f has no inverse function,

 

explain why not. (5 points)

 


 

5.

 


 

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 with the property that for all a in A g(f(a)) = a, and show that this

 

property holds. HINT: First identify the domain and the target of the function g. Second, think of g as

 

?undoing? what f does.

 


 

Chapter 4

 


 

Midterm (Show/Explain all Work)

 

6.

 


 

IST 230

 


 

Given the following circuit, complete the output column on the table below. (5 points)

 


 

x

 

1

 

1

 

1

 

1

 

0

 

0

 

0

 

0

 


 

Inputs

 

y

 

1

 

1

 

0

 

0

 

1

 

1

 

0

 

0

 


 

Output

 

z

 

1

 

0

 

1

 

0

 

1

 

0

 

1

 

0

 


 

Chapter 5

 

7. Draw the arrow diagram and the matrix representation for the following relation on the set {1, 2, 3, 4}.

 

R = { (1, 1), (1,4), (2,2), (3,4), (3, 2), (2, 1), (1, 3), (4,2)}.

 

a.

 


 

Draw the arrow diagram for R (a scan of a hand-drawn image is OK). (5 points)

 


 

Midterm (Show/Explain all Work)

 


 

IST 230

 


 

b. Draw the matrix representation for R. (5 points)

 


 

c.

 


 

8.

 


 

Explain why the relation R is or is not reflexive. (5 points)

 

You answer must begin with either ?The relation R is reflexive, because? or ?The relation R is not

 

reflexive, because?.

 


 

Define the following two relations on the set {a, b, c, d} (10 points)

 


 


 


 

S = { (a, b), (a, c), (c, d), (c, a) }

 

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

 


 

Write S ? R as a set of one or more ordered pairs.

 


 

Chapter 6

 

9.

 


 

Write an algorithm in pseudo-code with the following input and output (10 points).

 

Input: a1, a2,..., an, a sequence of integers.

 

Output: The smallest integer in the sequence.

 


 

10. What is the value of count after the last iteration of the outer for loop? Be sure show your work. (10

 

points)

 

count :=0

 


 

IST 230

 


 

Midterm (Show/Explain all Work)

 

For i= 1 to 2

 

For j=1 to 2

 

count :=j(count + 2i)

 

End-for

 

End-for

 


 

Chapter 7

 

11. Find the next three terms (terms

 


 

a2

 


 

Here, the right-hand side is j times the quantity (count + 2i)

 


 

a3

 


 

a 4 ) of the recursively defined sequence

 

a k =ak ?1 +3 k , for all integers k ? 2 , where a1=1 . (5 points)

 


 

 


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