I need help with these problems Please help Question 1 Let t

I need help with these problems!! Please help!

Question 1)

Let the following predicates be given. The domain consists of everything.

F(x) = x is a tool

H(x) = x is in the correct place

S(x) = x is in excellent condition

Express each of the following English sentences in terms of F(x), H(x), S(x), quantifiers, and logical connectives.

a) Something is not in the correct place.

b) All tools are in the correct place and are in excellent condition.

c) Nothing is in the correct place and is in excellent condition.

d) One of your tools is not in the correct place, but it is in excellent condition

Question 2

Let B(x), S(x), and A(x) be the predicates
B(x) : x is a Math major
S(x) : x is a CS major
A(x) : x is in Discrete Structure class

Translate each of the following quantified logic expressions ( provided in the file) into English considering the domain to consist of all people .

1. ?x(A(x))  ?(S(x)) ? (B(x)))
2. ¬?x(B(x))     

3. ?x( S(x) ? ¬B(x)) ? ¬A(x))

4. ? ¬(S(x)) ? ¬B(x))

Question 3

Negate each of the following statements:

1) If the teacher is absent, then some students do not complete their homework.

2) Some suspicions were substantiated.

3) No student in your class has taken a course in logic programming.

Question 4

Question 5

True or false: For the set of all negative integers, ?x (x + 1 < - x)

Question 6

Use a direct proof to show that the sum of two rational numbers is rational. (Recall that a number is rational if and only if it can be expressed as the ratio of integers.)

Question 7

Suppose x, y ? Z. Prove by contraposition that If x2 (y+3) is even, then x is even or y is odd.

Solution

1)

a) Something is not in the correct place.

xH(x)

b) All tools are in the correct place and are in excellent condition.

x(F(x)H(x)S(x))

c) Everything is in the correct place and in excellent condition.

x(H(x)S(x))

d) Nothing is in the correct place and is in excellent condition.

x(H(x)S(x))

e) One of your tools is not in the correct place, but it is in excellent condition.

x(F(x) (H(x)S(x)))

2) B(x) : x is a Math major
S(x) : x is a CS major
A(x) : x is in Discrete Structure class

1. x(A(x))  (S(x)) (B(x)))

One of your Discrete Structure class is not a CS major or it is a math major.

2. ¬x(B(x))

Everything is not a major math.

3. x( S(x) ¬B(x)) ¬A(x))

one of your CS major and it is not a math major or it is not in Discrete Structure class

4. ¬(S(x)) ¬B(x))

One of your subject in not a CS major and not a math major

6)

Proof.

Suppose r and s are rational numbers. [We must show that r + s is rational.] Then, by the definition of rational numbers, we have

                                            r = a/b    for some integers a and b with b 0.

                                            s = c/d   for some integers c and d with d 0.

So, by substitution, we have

                                            r + s = a/b + c/d

                                                    = (ad + bc)/bd

Now, let p = ad + bc and q = bd. Then, p and q are integers [because products and sums of integers are integers and because a, b, c and d are all integers. Also, q 0 by zero product property.] Hence,

r + s = p/q , where p and q are integers and q 0.

Therefore, by definition of a rational number, (r + s) is rational. This is what was to be shown.

And this completes the proof.

3)

1) If the teacher is absent,and none of the students do complete their homework.

2) All suspicions are not substantiated or None of the suspicions are not substantiated

3) All students in your class has taken a course in logic programming.