Question

1. Use a truth table to find if the following is valid or not valid: p → r q → r q ˅ ¬r Therefore, ¬p

Valid

Not Valid

Discrete Math

2. Indicate whether each expression is an equivalence of the following:

p ˄ q

   
p ˅ q

   
p → q

   
¬(p → q)

   
(p ˄ q) ˅ (p ˄ q)

   
¬ (¬p ˅ ¬q)

3. For the given values for p, q, and r, what is the result of the following proposition. A truth table might be helpful here.

¬(r → ¬q) ˅ (p ˄ ¬r)

   
p=T q=T r=T

   
p=T q=F r=T

   
p=T q=F r=F

   
p=F q=T r=T

   
p=F q=F r=F

QUESTION 5 For the statement P(x,y) meaning x+2y = xy, determine if the following are true or false.

   
P(0,0)

   
P(1,-1)

   
ƎyP(3,y)

   
ɄxP(x,2)

   
Ʉ x ƎyP(x,y)

QUESTION 6 Indicate whether each of the following is a proposition:

   
3 + 2 = 9

   
Get your raincoat

   
It is raining.

   
57

   
purple

QUESTION 7

For the premises "Someone in this class passed the exam", and "Everyone in this class read the textbook" imply the conclusion "Someone that read the textbook passed the exam". If C(x) is the predicate "x is in this class", R(x) is the predicate "x has read the textbook" and P(x) is the predicate "x passed the exam", what are the reasons each of the following steps can be deemed valid:

   
1. Ǝx(C(x) ˄ P(x))

   
2. C(a) ˄ P(a)

   
3. C(a)

   
4. Ʉx(C(x) → R(x))

   
5. C(a) → R(a)

   
6. R(a)

   
7. P(a)

   
8. P(a) ˄ R(a)

   
9. Ǝx(P(x) ˄ R(x))

   
10. C(a) ˄ R(a)

A. Premise

B. Addition

C.   DeMorgans Law

D.   Modus Tollens

E.   Modus Ponens

F.   Conjunction

G.   Existential Instantiation

H.   Existential Generalization

I.   Universal Instantiation

J.   Universal Generalization

K.   Simplification

QUESTION 8 Every proposition can be expressed in terms of AND, OR, and NOT. What proposition represents the following truth table?

       
p ˄ q

       
p ˅ q

       
p ↔ q

       
(p ˄ ¬q) ˅ (¬p ˄ q)

       
(¬p ˄ ¬q) ˅ (¬p ˄ ¬q)

QUESTION 9

Indicate whether the following are Tautologies or Contradictions:

   
p ˅ ¬p

   
p ˄ ¬p

   
p ˅ (q ˅ ¬q)

   
p ˄ (q ˄ ¬q)

   
((p → ¬q) ˄ q) → ¬p

QUESTION 10

For the premises "Everyone who read the textbook passed the exam", and "Ed read the textbook" imply the conclusion "Ed passed the exam". If R(x) is the predicate "x has read the textbook" and P(x) is the predicate "x passed the exam", what are the reason each of the following steps can be deemed valid:

   
1. Ʉx(R(x) → P(x))

   
2. R(Ed) → P(Ed)

   
3. R(Ed)

   
4. P(Ed)

   
5. R(Ed) ˄ P(Ed)

A.  
Premise

B.  
Conjunction

C.  
Addition

D.  
Simplification

E.  
Modus Ponens

F.  
Modus Tollens

G.  
DeMorgans Law

H.  
Universal Instantiation

I.  
Existential Instantiation

QUESTION 11 What is the negation of the following sentence:

It is Thursday and it is cold.

       
It is not Thursday and it is cold.

       
It is Thursday and it is not cold.

       
It is not Thursday and it is not cold.

       
It is not Thursday therefore it is not cold.

       
It is not Thursday or it is not cold.

QUESTION 14 Match the following statements with predicates and any necessary quantifiers:

   
Every Razzma is a Frat.

   
No Razzma is a Frat.

   
There is at least one Razma that is a Frat

   
There is a Razma that every Frat.

   
Some Razma are Green Frat.

A.  
R(x) ˄ F(x)

B.  
Ʉx(R(x) → F(x))

C.  
Ǝx(R(x) ˄ F(x))

D.  
Ʉx(R(x) ˄ F(x))

E.  
¬Ʉx(R(x) → F(x))

F.  
ƎyɄx(F(x) → R(x,y))

G.  
Ǝx(R(x) ˄ F(x) ˅ G(x))

H.  
ƎxƎy(R(x) ˄ F(y) ˄ G(y))

I.  
ƎxƎy(R(x) ˄ F(y) ˄ G(x,y))

J.  
ƎxƎy(R(x) → F(y) → G(x,y))

QUESTION 15 Indicate which of the following are equivalent to: A Yes B No

If Joe is going to the game, then I am going bowling.

   
Joe is going to the game if I am going bowling.

   
I am going bowling if Joe is going to the game.

   
Joe is going to the game implies I am going bowling.

   
Joe is going to the game whenever I am going bowling.

   
I am going bowling whenever Joe is going to the game.

Answer 1

1. p → r q → r q ˅ ¬r Therefore, ¬p
2. To form a truth table we form the each and every column as shown.
3. We know that T -> T and F -> F and F -> T are all true respectively and T -> F is false.

Truth table

	p	q	r	~r	p -> r	q -> r	q \/ ~r	~p
1	F	F	F	T	T	T	T	T
2	F	F	T	F	T	T	F	T
3	F	T	F	T	T	F	T	T
4	F	T	T	F	T	T	T	T
5	T	F	F	T	F	T	T	F
6	T	F	T	F	T	T	F	F
7	T	T	F	T	F	F	T	F
8	T	T	T	F	T	T	T	F

As we see the hypothesis premises row 2 and row 4 are same or all true, but the row 8 is not equivalent, as we see hypothesis premises are all true but the corresponding conclusion is false,

Which implies that the following is not valid.

According to Chegg's policy, I am answering the first full question only.

Thank you.