hw1math109

# hw1math109 - Similarly a< b and a is negative imply that...

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MATH 109 Assignment 1 Due: 4/4/11 #1.2(i) P Q P and Q not ( P and Q ) T T T F T F F T F T F T F F F T (ii) P Q not P not Q (not P ) or (not Q ) T T F F F T F F T T F T T F T F F T T T (iii) P Q not Q P and (not Q ) T T F F T F T T F T F F F F T F (iv) P Q not P (not P ) or Q T T F T T F F F F T T T F F T T

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#2.5 (i) P Q P Q not P not Q (not Q ) (not P ) T T T F F T T F F F T F F T T T F T F F T T T T Since the third and last columns are the same, the statements are equivalent. (ii) P Q P or Q not P (not P ) Q T T T F T T F T F T F T T T T F F F T F Since the third and last columns are the same, the statements are equivalent. #3.2 Proof. By deﬁnition, if ( a divides b ) and ( b divides c ) then there exist integers x, y Z such that ax = b and by = c . Hence a ( xy ) = ( ax ) y = by = c so a divides c . #3.6 Proof. By Axiom 3.1.2 (ii), if a < b and b is negative then b 2 < ab
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Unformatted text preview: . Similarly, a < b and a is negative imply that ab < a 2 . Hence b 2 < ab < a 2 as desired. Part I Problems #1: For the solution to the ﬁrst part of the problem, please see #2.5 (i). The contrapositive of the statement (i) ( f ( a ) = 0 ⇒ a > 0) is (vii): For real numbers a , if a is nonpositive then f ( a ) 6 = 0. The statements (iii) and (vi) are another pair of statements which are contrapositives of each other....
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hw1math109 - Similarly a< b and a is negative imply that...

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