# Exam 2 - R p,q if and only if mq = np(a(8 pts Prove that R...

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MATH 290 – 8 APRIL 2010 – EXAM 2 Answer all of the following questions on the answer sheets provided. Show all work, as partial credit may be given. This exam will be counted out of a total of 50 points. 1. (4 pts. each) For each n N , deﬁne the set B n by B n = { n, n + 1 , n + 2 , ..., n 2 } = { k N : n k n 2 } . (a) Prove that T n =1 B n = . (b) Prove that S n =1 B n = N . 2. (8 pts.) Use the Principle of Mathematical Induction to prove that for all n N , n X k =1 ( k · k !) = ( n + 1)! - 1 . 3. (8 pts.) Let a 1 = 3, a 2 = 9, and for n 3, a n = 5 a n - 1 - 6 a n - 2 . Use the Principle of Complete Induction to prove that for all n N , a n = 3 n . 4. Deﬁne the relation R on N × N by ( m,n
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Unformatted text preview: ) R ( p,q ) if and only if mq = np . (a) (8 pts.) Prove that R is an equivalence relation. (b) (3 pts.) List 3 elements of (3 , 4) /R . 5. Deﬁne the relation R on N × N by ( m,n ) S ( p,q ) if and only if m ≤ p and n ≤ q . (a) (8 pts.) Prove that S is anti-symmetric on N × N . (In fact, it is a partial order.) (b) (3 pts.) Is S a linear order ? Why or why not? 6. (8 pts.) For each a ∈ R , let A a = { ( x,y ) ∈ R × R : x + y = a } . Prove that for each a, b ∈ R , A a ∩ A b = ∅ or A a = A b ....
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