Predicate Logic and Quantifiers

1. You will create the domain for the next few questions using the unique digits in your id Example: StudentXs id is 23851123. They will list the digits in their id without duplicates: 1, 2, 3, 5, 8

What are the unique digits in your id?

Unique digits: 0,2,4,7,9

2. Let Q(x, y, z) denote x + y = z. The domain for x, y and z is the unique digits from your Emplid (Question1). Can you evaluate Q(1,1,2) based on the domain? If it can be evaluated, what is the truth value?

3. Let Q(x) denote x³ - x² > 0 . The domain for x is the unique digits from your Emplid (Question1). What is the truth value of xQ(x)? If it is false, give a counterexample.

4. Let Q(x) denote 2x = x. The domain for x is the unique digits from your Emplid (Question1). What is the truth value of xQ(x)? If it is true, give an example.

5. Let Q(x) denote x⁵ = 5^x. The domain for x is the unique digits from your Emplid (Question1). What is the truth value of xQ(x)? If it is true, give an example.

6. Let Q(x) denote (-6)^x 1. The domain for x is the unique digits from your Emplid (Question1). What is the truth value of xQ(x)? If it is false, give a counterexample.

7. Suppose that the domain of the propositional function P(x) consists of the integers 0, 1, 2, and 3. For each of the following quantifications, write an equivalent proposition using disjunctions ( ), conjunctions ( ), and negation ( )

a. xP(x)
b. xP(x)
c. xP(x)
d. xP(x)
e. xP(x)
f. xP(x)

8. A programming student wants to write a program that can be used to take information from a form and check whether a person is eligible for a vaccine for the students state.

Assume the program already has functions which can check the eligibility based on occupation (healthcare, first responder, grocery worker, etc ) and living situation (shelters, group living facilities, etc). The return value of the functions are assigned to boolean variables:

• eligibleOccupation
• eligibleLivingSituation

The student set up an if statement:

if(age < 65 && !eligibleOccupation && !eligibleLivingSituation )

cout<< "Not eligible for the vaccine";

else

//code for setting up appointment

A classmate wants to rewrite the code:

if( age > 65 && eligibleOccupation && eligibleLivingSituation )

//code for setting up appointment

else

cout<< "Not eligible for the vaccine";

Is the students classmates code equivalent? Write your suggestion for the boolean expression for the if statement. Identify the parts of the boolean expression that should be changed. How would you explain the changes to your classmate? Use your knowledge of propositional and/or predicate logic.

Rules of Inference

For the following questions, number and show all of the numbered steps in your argument with the reasons (premise/rule of inference/equivalence law) next to them.

1. From the premises: p (p q), s p. Show that q is a valid conclusion by providing the argument.

Steps

Reason

2. Given the premises p q, q r, r. Conclude (p r).

Steps

Reason

3. Given the premises p q and s p. Make a conclusion about whether s is true or false. Show the reason for your conclusion using argument form.

Steps

Reason

4. From the premises: r s, p q, q s. Show that (p v s) is a valid conclusion by providing the argument.

Steps

Reason