StudentFinalSolutions.pdf - RED Name Fall 2016 Student Submitted Sample Questions for the final Last Update December 8 2016 The questions are divided

StudentFinalSolutions.pdf - RED Name Fall 2016 Student...

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RED Name: Fall 2016 Student Submitted Sample Questions for the final Last Update: December 8, 2016 The questions are divided into three sections: True-false, Multiple Choice, and Written Answer. I will add questions to the end of each section, so the number of a question should never change–you can refer to “T/F question 5” no matter when you download the file (as long as it is after five questions have been put in). 1. True-False Questions 1. If we expand the expression (3 x + y ) 6 , the coefficient of the x 5 y term is 1458. Proof. True. The binomial coefficient of the x 5 y term is ( 6 1 ) , which equals 6. 6 * 3 5 * 1 1 = 1458. 2. P ( P = Q ) is a tautology. Proof. False. Consider the truth table: P Q P = Q P ( P = Q ) T T T T T F F F F T T F F F T F Therefore, P ( P = Q ) is not a tautology. 3. The statements P = Q and ”For P, it is necessary that Q” have the same meaning. Proof. True 4. If A × B = , then A = and B = . Proof. False. ”Then A = or B = ” would make the statement correct. 5. The following statements are logically equivalent: P ( Q R ) ( P Q ) ( P R ). Proof. These statements are logically equivalent and can be shown to be logically equivalent through a truth table. 6. Let A be the set A = { 1 , 3 , 7 , 20 , 400 , 5000 } . The size of the P ( A ) = 36. Proof. False. The size of the power set is 2 | A | , not | A | 2 . 7. If P and Q are statements, then ( P ) = ( P = Q ) is a tautology. Proof. True. Since ( P = Q ) is true except when P is true and Q is false, this is the only case we have to check, since otherwise, the truth value of the ”if” statement does not matter. In this case, P is true, making P false. Since both the ”if” and ”then” statements are false, this is a tautology, since it will always be true, regardless of the truth values of P or Q . 8. Any element x Q can be written as a b , where a, b N . 1
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2 Proof. False. This is true for a, b Z , but since neither a nor b can be negative, neither can x , and some negative numbers are elements of Q . Note that 0 cannot be a solution to a b either. 9. ( P ( P )) is logically equivalent to P ( P Q ). Proof. True. P Q P P ( P ) ( P ( P )) ( P Q ) P ( P Q ) T T F F T T T T F F F T F T F T T F T T T F F T F T T T 10. “ c Z , b N , c 2 6 = b ” is the negation of “ b N , c Z , c 2 = b ”. Proof. False. The negation is b N , c Z , c 2 6 = b . 11. The number of length 5 lists that can be created from the set { A,B,C,D,E,F,G,H } is 8 5 . Proof. True. For each element in a list of length 5 (with repetition permitted), there are 8 choices. By the Multiplication Principle, there are 8 5 lists of length 5. 12. 2 ⊆ { 2 , { 2 }} . Proof. False. 2 is not a set. 13. x Z , y Z , x = y + 2 Proof. False. For any integer y , there is only one integer x such that x = y + 2. That x value is unique for each y value because this is a bijection. Therefore there is no single x for which y + 2 = x is true for all y . 14. Let x N . 3 x being odd implies x is odd Proof. True. Using proof by the contra-positive.
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  • Spring '17
  • Dr. David Tweedle
  • Natural number, Prime number, Euclidean algorithm, Q p

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