day1-sol - Review Problems 1 ICME and MS&E Refresher...

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Unformatted text preview: Review Problems 1 ICME and MS&E Refresher Course September 19, 2011 Warm-up problems 1. For the following matrices A = 1 2 1 2 1 2 1 2 B = 1 − 1 C = AB = 1 2 − 1 2 1 2 − 1 2 find all powers A 2 ,A 3 , (which is A 2 times A ), ... and B 2 ,B 3 ,... and C 2 ,C 3 ,... . Solution: It is easy to verify that (a) A = A 2 = A 3 ... (b) B 2 = B 4 = ··· = bracketleftbigg 1 1 bracketrightbigg B = B 3 = B 5 = ... (c) C 2 = C 3 = ··· = bracketleftbigg bracketrightbigg 2. Use Venn diagram to prove the following distributive law of sets: A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) . Solution: For a good illustrative solution, please visit the following website: http://pirate.shu.edu/~wachsmut/ira/logic/proofs/distlaw.html 3. Which of the following functions are injective (one-one), surjective (onto), or bijective (one-one and onto)? The domain for all functions is the entire real line R : f ( x ) = x − 1 , g ( x ) = cos( x ) , h ( x ) = x 2 − 4 . 4. Let A and B be two sets. We say that A and B have the same cardinality if there is a bijective function f from A to B . We denote this as card( A ) = card( B ) . If there exists a function f from A to B that is injective, what can you say about card( A ) and card( B ) . What if f is surjective? Solution: If f is surjective, card( A ) ≤ card( B ), if f is surjective the inequality is reversed. 5. A sequence { x n } ∞ n =1 is called monotone increasing if x n +1 ≥ x n for all n. A monotone decreasing sequence is defined similarly. Comment whether the following sequences are monotone increasing or decreasing: braceleftbigg 1 n bracerightbigg ∞ n =1 , braceleftbigg n n + 1 bracerightbigg ∞ n =1 . Solution: 1 (a) x n x n +1 = n n + 1 ≤ 1 therefore, x n ≤ x n +1 and this sequence is monotonic decreasing. (b) x n − x n +1 = n n + 1 − n + 1 n + 2 = 1 n ( n + 2) ≥ therefore, x n ≥ x n +1 and the sequence is monotonic increasing. 6. Deviation of middle element value from average : Suppose x is an n-vector, with n = 2 m − 1 and m ≥ 1 . We define the middle element value of x as x m . Define: f ( x ) = x m − 1 n n summationdisplay i =1 x i , which is the difference between the middle element value and the average of the coefficients in x ....
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