Category: Discrete Math

2.1
Write the statements in symbolic form using the symbols ∼,
∨,
and ∧
and the

2.1
Write the statements in symbolic form using the symbols ∼,
∨,
and ∧
and the indicated letters to represent component statements.
6. Let s = “stocks are increasing” and i = “interest rates
are steady.”
a. Stocks are increasing but interest rates are steady.
b. Neither are stocks increasing nor are interest rates
steady.
8. Let h = “John is healthy,” w = “John is wealthy,” and s =
“John is wise.”
a. John is healthy and wealthy but not wise.
b. John is not wealthy but he is healthy and wise.
c. John is neither healthy, wealthy, nor wise.
d. John is neither wealthy nor wise, but he is healthy.
e. John is wealthy, but he is not both healthy and wise.
Determine whether the statement forms in 16-24 are logically
equivalent. In each case, construct a truth table and include a sentence
justifying your answer. Your sentence should show that you understand the
meaning of logical equivalence.
22. p ∧ (q ∨ r) and (p ∧ q) ∨
(p ∧
r)
Use De Morgan’s laws to write negations for the statements
in 25-31.
26. Sam is an orange belt and Kate is a red belt.
28. The units digit of 467 is 4 or it is 6.
determine whether the statements in (a) and (b) are
logically equivalent.
45. a. Bob is a double math and computer science major and
Ann is a math major, but Ann is not a double math and computer science major.
b. It is not the case that both Bob and Ann are double math
and computer science majors, but it is the case that Ann is a math major and
Bob is a double math and computer science major.
2.2
Construct truth tables for the statement forms in this
question
6. (p ∨ q) ∨ (∼p ∧ q) → q
11. (p → (q → r)) ↔ ((p ∧ q) → r)
14.
write each of the two statements in symbolic form and
determine whether they are logically equivalent. Include a truth table and a
few words of explanation.
16. If you paid full price, you didn’t buy it at Crown
Books. You didn’t buy it at Crown Books or you paid full price.
20. Write negations for each of the following statements.
(Assume that all variables represent fixed quantities or entities, as
appropriate.)
a.If P is a square, then P is a rectangle.
b.If today is New Year’s Eve, then tomorrow is January.
c.If the decimal expansion of r is terminating, then r is
rational.
d.If n is prime, then n is odd or n is 2.
e.If x is nonnegative, then x is positive or x is 0.
f.If Tom is Ann’s father, then Jim is her uncle and Sue is
her aunt.
g.If n is divisible by 6, then n is divisible by 2 and n is
divisible by 3.
Use truth tables to establish the truth of each statement in
this question
24. A conditional statement is not logically equivalent to
its converse.
27. The converse and inverse of a conditional statement are
logically equivalent to each other.
Rewrite the statements in if-then form.
37. Payment will be made on the fifth unless a new hearing
is granted.
Use the
contrapositive to rewrite the statements in this question in if-then form in
two ways.
43. Doing homework regularly is a necessary condition for
Jim to pass the course.
Use the logical equivalences p →q ≡∼p ∨
q and p ↔ q ≡ (∼p
∨
q) ∧
(∼q
∨
p) to rewrite the given statement forms without using the symbol → or ↔, and (b) use the logical
equivalence p ∨ q ≡∼(∼p
∧
∼q)
to rewrite each statement form using only ∧ and ∼.
49. (p →r) ↔ (q →r)
2.3
Use modus ponens or modus tollens to fill in the blanks in
the arguments of 1-5 so as to produce valid inferences.
1.If √2 is rational, then √2=a/b for some integers a and b.
It is not true that √2=a/b for some integers a and b.
__________________________________________________.
2. If 1 – 0.99999 … is less than every positive real number,
then it equals zero.
___________________________________________________________________________
The number 1 – 0.99999 … equals zero.
3. If logic is easy, then I am a monkey’s uncle. I am not a
monkey’s uncle.
__________________________________________________.
4. If this figure is a quadrilateral, then the sum of its
interior angles is 360°.
The sum of the interior angles of this figure is not 360°.
__________________________________________________.
5. If they were unsure of the address, then they would have
telephoned.
__________________________________________________.
They were sure of the address.
Use truth tables to determine whether the argument forms in this
question are valid. Indicate which columns represent the premises and which
represent the conclusion, and include a sentence explaining how the truth table
supports your answer. Your explanation should show that you understand what it
means for a form of argument to be valid or invalid.
6. P → q
q → p
p ∨ q
7.
P
P         →    q
∼  q  ∨   r
R
8.          p  ∨  q
P   →    ∼      q
P     →      
r
r
9.          p ∧
q → ∼
r
P
∨
∼
q
∼
p    → p
∼
r
Use symbols to write the logical form of each argument in this
question, and then use a truth table to test the argument for validity.
Indicate which columns represent the premises and which represent the
conclusion, and include a few words of explanation showing that you understand
the meaning of validity.
22. If Tom is not on team A, then Hua is on team B.
If Hua is not on team B, then Tom is on team A.
Tom is not on team A or Hua is not on team B.

1. If a set B has n elements, then what is the total number of subsets of B. Jus

1. If a set B has n elements, then what is the total number of subsets of B. Justify your answer.
Solution: If a set B has “n” elements, then the total number of subsets of B is 2n.
For example, if B contains 5 elements, say B = {1, 2, 3, 4, 5}, then the total number of subsets of B is 25 = 32.
2. If X and y are the two finite sets, such that n(X U Y) = 36, n(X) = 20, n(Y) = 28, then find n( X ∩ Y).
Solution: n(X) = 20, n (Y) = 28, n (X U Y) = 36.
As we that, n(X U Y) = n(X) + n(Y) – n(X ∩ Y)
On rearranging the above formula, we get;
n(X ∩ Y) = n(X) + n(Y) – n(X U Y)
Now, substitute the given values in the above formula, we get;
n(X ∩ Y) = 20 + 28 – 36
n(X ∩ Y) = 48 – 36
n(X ∩ Y) = 12.
Hence, n(X ∩ Y) is 12.

Here is the list of 6 problems, you need to solve each and every problems withou

Here is the list of 6 problems, you need to solve each and every problems without skipping any steps.

All of these questions must show all steps on how you got the answer. Thank you.

All of these questions must show all steps on how you got the answer. Thank you. There are 6 problem total. I will send all when bid is accepted

I have attached the assignment instructions below. I have also attached the cour

I have attached the assignment instructions below. I have also attached the course lecture materials and textbook for review.
Textbook: https://github.com/phoenixsense/CS500/blob/master/…
The assignment needs to be done in Word document format
The deadline is 19 hours from now
I need two versions to submit, same correct answers but a different structure so that it looks different.

I have attached the assignment instructions below. I will also attach the course

I have attached the assignment instructions below. I will also attach the course lecture materials for review.
Two versions need to be submitted, same correct answers but different structure
The assignment needs to be done in Word document format

I have attached the assignment instructions below. I will also attach the course

I have attached the assignment instructions below. I will also attach the course lecture materials for review.
Two versions need to be submitted, same correct answers but different structure
The assignment needs to be done in Word document format

I have attached the assignment instructions below. I can also attach the course

I have attached the assignment instructions below. I can also attach the course lecture materials for review if needed.
Word document needs to be used and converted to a PDF after for submission

I have attached the assignment instructions below. I can also attach the course

I have attached the assignment instructions below. I can also attach the course lecture materials for review if needed.
Word document needs to be used and converted to a PDF after for submission

context for questions:
Mastermind is a game in which you have to guess a hidden

context for questions:
Mastermind is a game in which you have to guess a hidden code made of 4 “color pegs” (a tuple), from a set of 7 colors {R,G,B,Y,W,K,O}. The code maker responds to your guess with smaller “key pegs,” which are black or white (a multiset) and indicate exact or color matches with the code. (More precise definition below.)
several students were asked to answer/fix the chat gpt mistakes of questions. an example is listed below. This only serves to give context for the questions in the pdf. there were a bunch of questions like this, but this is one example.
question #1.
You are playing Mastermind by making a guess that is a 4-tuple drawn from a set of 7 color pegs, {R,B,G,Y,W,K,O} (order matters, repeats allowed). The codemaker responds with a multiset 0 to 4 black or white key pegs (order doesn’t matter; repeats allowed). Each b indicates some guess peg matches the code in position and color; w indicates some guess peg has a right color in the wrong position. (A more precise statement of the rules is in CW#30.)
You want to know how many codes are consistent with the response to your guess, because that is the first step to narrowing this down to one code.
Here is your guess and the response:
RRRYww
https://chat.openai.com/share/ea81a6de-48cd-442f-8…