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347e1pracSpring05

# 347e1pracSpring05 - (a For all functions f R → R if f is...

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Math 347 Exam #1 Spring 2005 Name All problems except # 1 require proofs and/or explanations. 1. (16 points) Write the negation of each statement. (a) x R , if x 2 > x , then x < 0 or x > 1. Negation: (b) ε > 0, N N such that for each integer n N , | a n - L | < ε . Negation: 2. (12 points) Let S = { x R : | x + 1 | > x } . Prove that S = R . 3. (24 points) (a) Let n Z . Prove that n is even if and only if 3 n - 2 is even. (b) Show that if f : R R is increasing, then g ( x ) = - f ( - x ) is also increasing. 4. (16 points) Prove that for all n N , n X j =1 j ( j + 1) = n ( n + 1)( n + 2) 3 . 5. (22 points) For each statement, determine whether it is true or false and give a proof.
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Unformatted text preview: (a) For all functions f : R → R , if f is bounded, then h ( x ) = ( f ( x )) 2 is bounded. (b) For all sets A and B , if A-B = ∅ , then A = B . 6. (10 points) Suppose P (1) ,P (2) ,P (3) ,... is an inﬁnite sequence of statements. Also, suppose that for all n ∈ N , if P ( n ) and P ( n + 1), then P ( n + 5) (think of this as an inductive step). What statements need to be proved in a base step in order to conclude that P ( n ) is true for all n ? Explain your reasoning....
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