midterm-soln-s05 - Math 135 - Midterm Examination 1 Math...

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Math 135 - Midterm Examination 1 Math 135 Algebra Midterm Solutions Question 1. (a) [4 marks] Consider the following two statements: x y ( P ( x ) = Q ( y )) , and ( x P ( x )) = ( y Q ( y )) Give an example of a universe of discourse, and meaning for P and Q for which the frst statement is true . Give an example where the second statement is False . Solution: There are many possible correct answers to this question. Here’s one: For the positive integers, and P ( x ) = Q ( x ) = “ x < 1”, the ±rst statement is true. For the positive integers, P ( x ) = “ x 1”, and Q ( y ) = “ y < 1”, the second statement is false. (b) [4 marks] Negate the two statements given in part (a) above, and simplify them so that no quanti±er or compound statement is negated. Show the steps in the simpli±cation. Solution: ¬ ( x y ( P ( x ) = Q ( y ))) is equivalent to x y ¬ ( P ( x ) = Q ( y )) x y ( P ( x )) ∧ ¬ Q ( y )) . ¬ (( x P ( x )) = ( y Q ( y ))) is equivalent to ( x P ( x )) ∧ ¬ ( y Q ( y )) ( x P ( x )) ( y ¬ Q ( y )) . (c) [4 marks] Are the two statements given in part (a) above equivalent? Justify your answer appropriately. Solution: The two statements are equivalent. There are several ways of proving this. The easiest is to prove that the negations of the two statements we obtained in part (b) above are equivalent. We prove this below. Let S 1 be the negation of the ±rst statement, S 2 that of the second. Suppose S 1 is true. Then, for every pair of elements x, y in the universe, P ( x ) is true and Q ( y ) is false. This means that x P ( x ) is true, and y ( ¬ Q ( y )) is also true. This implies that S 2 is true. Conversely, suppose that S 2 is true. Then, the two statements x P ( x ) and y ( ¬ Q ( y )) are both true. Thus, for every pair x, y in the universe, P ( x ) ∧ ¬ Q ( y ) is true. In other words, S 1 is true.
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Math 135 - Midterm Examination 2 Question 2. (a) [4 marks] Factorize the two integers 11571 and 9338 into a product of primes, and compute their greatest common divisor. Explain the method you used to factorize the numbers, and show your steps.
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midterm-soln-s05 - Math 135 - Midterm Examination 1 Math...

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