319ISOL - MAT 319 SPRING 2008 MIDTERM I WRITE YOUR NAME...

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MAT 319 SPRING 2008 MIDTERM I !!! WRITE YOUR NAME, STUDENT ID BELOW !!! NAME : ID : THERE ARE SIX (6) PROBLEMS. THEY HAVE THE INDICATED VALUE. SHOW YOUR WORK!!! DO NOT TEAR-OFF ANY PAGE NO CALCULATORS NO PHONES ON YOUR DESK: ONLY test, pen, pencil eraser. 1 40pts 2 40pts 3 40pts 4 50pts 5 40pts 6 40pts Total 250pts 1
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2 !!! WRITE YOUR NAME, STUDENT ID AND LECTURE N. BELOW !!! NAME : ID : LECTURE N. 1. Prove that the number 3 12 is irrational. Solution. By contradiction, assume 3 12 = r/s , where r, s are coprime natural numbers. We get 12 s 3 = r 3 . Then 2 divides r, so that r = 2 r 1 . We get 2 2 · 3 s 3 = 2 3 r 3 1 . We get 3 s 3 = 2 r 3 1 , so that 2 divides s. This contradicts r, s coprime.
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3 2. In each case provide an example or state that it is not possible to do so and explain why. 2a Give one example of a countable set that is not finite. 2b Give one example of a countable set that is not denumerable. 2c Give one example of an uncountable set. 2d Give one example of an uncountable set that is not infinite.
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