Math 3000 Sample Exam I

There are seven (7) questions listed below, you may choose to do any five (5) of them. Each question will count 20 points. If you complete more than 5 questions, you must clearly indicate which 5 you wish to be graded. There will be no extra credit - don't bother asking for it. Good Luck!!.
1. Use a truth table to show that the two statements (P Q) R and (P (Q R)) are logically equivalent.

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2. Use logical symbols to rewrite these statements and determine if the argument is valid (that is, the premise [first three statements] implies the conclusion [last statement]).

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3. If P(x) and Q(x) are propositional functions involving x, show that

(x)(P(x)Q(x))((x)P(x) (x)Q(x))

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4. Prove by induction that for all natural numbers n, we have:

2 + 22 + 23 + ... + 2n = 2n+1 - 2.

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5. Determine whether the following statements are true or false. If false, briefly explain why?

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6. Let A and B be subsets of a universal set U. Prove that:

(A - B)(B - A) = (AB) - (AB)

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7. Prove that there are an infinite number of primes.

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Answer to question 1:

PQRPQ(PQ) RQRP(QR)
TTTTTTT
TTFTFFF
TFTFTTT
TFFFTTT
FTTFTTT
FTFFTFT
FFTFTTT
FFFFTTT
Since the fifth and seventh columns are identical, the two expressions are logically equivalent.

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Answer to question 2:

Let S = "turtles can sing", A = "artichokes can fly" and D = "dogs can play chess". The given statements can now be written symbolically as:

SA,
A(S~D),
DS.
By transitivity, the first two implications give S(S~D). But, since S and D are logically equivalent, this means that S(D~D), which is a contradiction. Since a true statement cannot imply a contradiction, we must have that S is false, i.e., ~S is true - turtles can not sing.

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Answer to question 3:

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Answer to question 4:

Let A be the set of all natural numbers for which the statement is true.
The statement, with n = 1, is 2 = 22 - 2. As the RHS evalutes to 2, the statement is true in this case, and so, 1 is in A. Now, assume that the statement is true for the integer m, and consider n = m + 1. We have:
21 + 22 + 23 + ... + 2m + 2m+1 = 2m+1 - 2 + 2m+1 = 2(2m+1) - 2 = 2m+2 -2, showing that the statement is true for n = m + 1. Thus, A is an inductive set and by PMI the statement is true for all n.

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Answer to question 5:

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Answer to question 6:

x(A - B)(B - A)((xA)(xB))((xB)(xA))
(((xA)(xB))(xB))(((xA)(xB))(xA))
(((xA)(xB))((xB)(xB)))(((xA)(xA))((xB)(xA)))
(((xA)(xB))((xB)(xA)))
(xAB)(xBA)
x(AB) - (BA)

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Answer to question 7:

See returned Special Assignment #1.

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