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MATH 135 Final Exam
Page 2 of 15
Name:
[6] 1. (a) Determine the complete solution to the Linear Diophantine Equation 767x + 472y = 7847
[4]
(b) Determine all non-negative integer solutions to the Linear Diophantine Equation 767x + 472y = 7847
MA
MATH 135
Midterm #1 Solutions
Winter 2005
[6] 1. Use truth tables to show that the following statements are equivalent: (P Q) AND (R P ) and (P AND Q) OR NOT (P OR R). Solution: P T T T T F F F F P T T T T F F F F Q R T T T F F T F F T T T F F T F F Q R T
Some Highlights of Math 135 - Blocks V - VIII 1. How do you test whether an integer is divisible by 9? See whether the sum of its digits is divisible by 9. 2. How do you test whether an integer is divisible by 11? See whether the alternating sum of its di
Some Highlights of Math 135 - Blocks I - IV 1. NOT (x, P (x) is equivalent to . (x, NOT P (x) 2. NOT (x, P (X) is equivalent to . (x, NOT P (x) 3. Contrapositive Law: P Q is equivalent to. NOT Q NOT P . 4. Describe Proof by Contradiction for proving P . A
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MATH 135 Final Exam
Page 2 of 13
Name:
[6] 1. (a) Determine the complete solution to the linear Diophantine equation 391x + 253y = 2760.
[4]
(b) Suppose that the complete solution to a linear Diophantine equation is given by j = -23 - 3n k = 521
Some Highlights of Math 135 1. NOT (x, P (x) is equivalent to . (x, NOT P (x) 2. NOT (x, P (X) is equivalent to . (x, NOT P (x) 3. Contrapositive Law: P Q is equivalent to. NOT Q NOT P . 4. Describe Proof by Contradiction for proving P . Assume that P is
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2001 midterm
Time: 90 mins1 1. (a) Use the truth table to prove the Contrapositive Law: P = Q is equivalent to NOT Q = NOT P. (b) If the universe of discourse is the set of real numbers, what does the following statement mean in English? Give reasons. x y