Write out a detailed proof use the same figure ba c

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Unformatted text preview: d X C which means that every member of X must fail to belong to at least one of the sets A and B and C. Thus no member of X can belong to A B C. Since the sets A and B and C are subsets of X we also know that no object x lying outside the set X can belong to A B C. Therefore A B C  . 8. Given sets A, B and C, determine which of the following identities are true. Hint: The truth values are given here. Explain carefully why they are correct. a. A B CA B A C True b. A Þ B C  AÞB A Þ C False c. A Þ B C  AÞB A C False d. A Þ B C  AÞB A C False e. A B CA B f. A B CA B A C False g. A B CA B A C False h. A B CA BÞA C False i. A B CA BÞA C True j. A B CA B k. A  A BÞA B True C False A Þ C False l. p A Þ B  p A Þ p B False m. p A B pA n. A BÞC  A o. A B p. A B p B True BÞA C True CA B A C True CA B A C True 9. Is it true that if A and B are sets and A  A B then the sets A and B are disjoint from each other? Hint: Yes 10. Given that A is a set with ten members, B is a set with seven members and that the set A members, how many members does the set A Þ B have? 31 B has four 11. Give an example of a set A that contains at least three members and that satisfies the condition A pA. Hint: Define A 12. For which sets A do we have A , , pA? Solution: This happens for all sets. Every set is a subset of itself. Exercises on Functions 1. Given that f x  x 2 for every real number x, simplify the following expressions: a. f 0, 3 We have f 0, 3  0, 9 . f 2, 3  0, 9 . 1 3, 4  b. f 2, 3 We have c. f 1 3, 4 We have f 2. 2, 2 . Point at the equation f x  x 2 and then click on the button in your computing toolbar. Then work out the the expressions in parts a and b of the preceding exercise by pointing at them and clicking on the evaluate button. 3. Supply each of the definitions f x  x 2 and g x  2 3x to Scientific Notebook and then ask Scientific Notebook to solve the equation f g x  g f x. 4. Supply the definition fx  x 2 1 2x to Scientific Notebook. In this exercise we shall see how to evaluate the composition of the function f with itself up to 20 times starting at a variety of numbers. Open the Maple menu, click on Calculus and move to the right and select Iterate. In the iterate dialogue box fill in the function as f, the starting value as 3 and the number of iterations as 20. Repeat this process with different starting values. Can you draw a conclusion from what you see? 32 5. Given that f x  x 2 for all x a. R and g x  1  x for all x R, simplify the following expressions: f g 0, 1 We have fg 6.  f 1, 2  1, 4 . 0, 1  g f 0, 1  g 0, 1  1, 2 . gg c.  f g 0, 1 gf b. 0, 1 0, 1  g g 0, 1  g 1, 2  2, 3 . g f 0, 1 We have g g 0, 1 We have a. Given that f x  3x 2 / x  1 for all x its range. Given any number y, the equation R 1 , determine whether or not f is one-one and find y  3x 2 x1 holds when x  1 y  3x 2 which can be expressed as x 3 y  2  y. In the event that y  3 the lattter equation says that 0  5 and is th...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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