# Math109Fall2019HW1SolutionsRubric.pdf - Homework 1...

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Homework 1 Solutions and Rubric Point distribution (out of 20 points in total): Problem 2.5(ii) – 3 points. Problem 3.1 – 4 points. Problem I.4 – 4 points. Rolle’s Theorem – 4 points. Completeness – 5 points. This is for attempting the remaining 14 problems. See the back of the book for the remaining solutions. Problems I 1. By using truth tables prove that, for all statements P and Q , the statement ‘ P Q 0 and its contrapositive ‘(not Q ) (not P ) 0 are equivalent. In Example 1.2.3 identify which statement is the contrapositive of the statement (i) ( f ( a ) = 0 a > 0). Find another pair of statements in that list which are contrapositives of each other. Since the last columns of the truth tables for P Q and ¬ Q ⇒ ¬ P are the same, the statements are equivalent: P Q P Q T T T T F F F T T F F T P Q ¬ Q ¬ P ¬ Q ⇒ ¬ P T T F F T T F T F F F T F T T F F T T T (i) f ( a ) = 0 a > 0 is equivalent to its contrapositive (vii) a 6 > 0 f ( a ) 6 = 0.

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