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Exam 3 - Solutions MAT 300(H Thieme 1 30 2 23 3 20 4 10 5...

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Solutions MAT 300 ( H. Thieme ) Test 3; April 21, 2006 1 2 3 4 5 30 23 20 10 17 100 Work your problems in the space provided. Show all work clearly. 1. Let A = { 1 , 2 , 3 } and consider the following relation R on A , R = { (1 , 2) , (2 , 1) , (3 , 2) } . [30 points] (a) Is R a function? [3 points] Yes, for every first component there is at most one second component. (b) Dom( R )= { 1 , 2 , 3 } [3 points] (c) Is R : A A ? Explain. [3 points] Yes, because Dom( R ) = A . (d) Ran( R )= { 1 , 2 } [3 points] (e) Is R surjective (onto)? Explain. [3 points] No, because Ran( R ) 6 = A . (f) Is R injective (one-to-one)? Explain. [3 points] No, because (1 , 2) , (3 , 2) R . (g) R - 1 = { (1 , 2) , (2 , 1) , (2 , 3) } . [3 points] (h) Is R - 1 a function? Explain. [3 points] No, because (2 , 1) , (2 , 3) R - 1 . (i) R - 1 ( { 1 , 2 } ) = { 1 , 2 , 3 } [3 points] (j) R R = { (1 , 1) , (2 , 2) , (3 , 1) } [3 points]
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Name page 2 2. Let A and B be sets and f : A B , g : B C . Show (a) If f and g are surjective, so is g f . [10 points] (b) If f and g are injective, so is g f . [10 points] (c) If f and g are bijective, so is g f . [3 points] See the book, Theorem 3.4.1.
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Name page 3 3. Consider R , the real numbers, with the usual order . Define a relation
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