Predicates P, T and E are defined below. The domain of discourse is the set of a

Question

Predicates P, T and E are defined below. The domain of discourse is the set of all positive integers.

P(x): x is even

T(x, y): 2x = y

E(x, y, z): xy = z

Indicate whether each logical expression is a proposition. If the expression is a proposition, then give its truth value.

(h)

T(5, 16) E(6, 3, 36)

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In this problem, the domain of discourse is the set of all integers. Which statements are true? If an existential statement is true, give an example. If a universal statement is false, give a counterexample.

(e)

x (x2 > 0)

(f)

x (x2 > 0)

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The domain for this problem is a set {a, b, c, d}. The table below shows the value of three predicates for each of the elements in the domain. For example, Q(b) is false because the truth value in row b, column Q is F.

Which statements are true? Justify your answer.

(e)

x R(x)

(f)

x R(x)

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In the following question, the domain of discourse is a set of male patients in a clinical study. Define the following predicates:

P(x): x was given the placebo

D(x): x was given the medication

M(x): x had migraines

Translate each statement into a logical expression. Then negate the expression by adding a negation operation to the beginning of the expression. Apply De Morgan's law until each negation operation applies directly to a predicate and then translate the logical expression back into English.

Sample question: Some patient was given the placebo and the medication.

x (P(x) D(x))

Negation: ¬x (P(x) D(x))

Applying De Morgan's law: x (¬P(x) ¬D(x))

English: Every patient was either not given the placebo or not given the medication (or both).

(c)

There is a patient who took the medication and had migraines.

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Write the negation of each of the following logical expressions so that all negations immediately precede predicates. In some cases, it may be necessary to apply one or more laws of propositional logic.

(d)

x y (P(x, y) P(y, x))

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Which of the following arguments are invalid and which are valid? Prove your answer by replacing each proposition with a variable to obtain the form of the argument. Then prove that the form is valid or invalid.

If 22 is an irrational number, then 2222 is an irrational number.

2222 is an irrational number.

22 is an irrational number.

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Prove that each argument is valid by replacing each proposition with a variable to obtain the form of the argument. Then use the rules of inference to prove that the form is valid.

(a)

If I drive on the freeway, I will see the fire.

I will drive on the freeway or take surface streets (or both).

I am not going to take surface streets.

I will see the fire.

P Q R a T T F b T F F c T F F d T F F

Explanation / Answer

(According to Chegg policy, only four subquestions will be answered. Please post the remaining in another question)

P(x): x is even

T(x, y): 2x = y

E(x, y, z): xy = z

(h) T(5, 16) E(6, 3, 36) : This is a proposition.

T(5,16) is false as 2*5 = 10 and not 16.

E(6,3,36) is false as 6*3 = 18 and not 36.

As false -> false is true, the statement has truth value T.

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(e)

x (x2 > 0)

This statement is false as for x = 0, x2 = 0

(f)

x (x2 > 0)

This statement is true as for x = 1, x2 = 1 > 0

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(e)

x R(x)

This statement is false as R(x) is false for all values a,b,c,d.

(f)

x R(x)

This statement is false as R(x) is false for all values a,b,c,d.

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P(x): x was given the placebo

D(x): x was given the medication

M(x): x had migraines

(c) There is a patient who took the medication and had migraines.

x (D(x) ^ M(x))

Negation: ¬x (D(x) ^ M(x))

Applying De-Morgan's law: x (¬D(x) ¬M(x))

English: Every patient either did not take medication or did not have migranes.

P Q R a T T F b T F F c T F F d T F F

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