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Unformatted text preview: 1 MATH 135 F 2011: Assignment 3 Due: 8:30 AM, Wednesday, 2011 Oct. 12 in the dropboxes outside MC 4066 Write your answers in the space provided. This assignment is structurally similar to the midterm. The rst ve questions of the midterm will be just like the rst ve questions of this assignment, except for dierent statements or numbers. The remaining questions on both the midterm and the assignment ask you to read or discover proofs. Because assignments have more time than the midterm, this assignment is longer and harder than the midterm. Because of the relative length of this assignment, only a subset of questions will be marked. Solutions to all of the questions will be posted. Cite any proposition or denition you use. Family Name: Solutions First Name: I.D. Number: Section: Mark: (For the marker only.) If you used any references beyond the course text and lectures (such as other texts, discussions with colleagues or online resources), indicate this information in the space below. If you did not use any aids, state this in the space provided. 2 1. The truth table for if and only if ( ) is given below. A B A B T T T T F F F T F F F T Construct a truth table for A B C ( A B ) ( A C ). A B C B C A B C A B A C T T T T T T T T T T T F T T T F T T T F T T T F T T T T F F F F F F F T F T T T T T T T T F T F T T T T T T F F T T T T T T T F F F F T T T T T 2. What is the complete solution to the linear Diophantine equation 1950 x-770 y = 30 ? We begin with the EEA applied to 1950 and 770. We will adjust the signs later. x y r q 1 1950 1 770 1-2 410 2-1 3 360 1 2-5 50 1-15 38 10 7 77 195 5 Now we see that 1950(-15)-770(-38) = 10 Multiplying by 3 gives one particular solution, x =-45, y =-114 to 1950 x-770 y = 30. The complete solution is x =-45-77 n y =-114-195 n for n Z ....
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