Other 2 suppose that is the relation in r 2 defined

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Unformatted text preview: y, x belong to S S we do not have x  y. 7. Prove that if r 1 and r 2 are partial orders in a set S then r 1 r 2 is also a partial order in S. What about r 1 Þ r 2 ? It is easy to see that if r 1 and r 2 are partial orders of S then so is r 1 r 2 . However r 1 Þ r 2 can fail. For example, both and are partial orders of R but the union of these two relations is R R which is not a partial order of R. 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 b. f 2, 3 We have f c. f 1 2, 3  0, 9 .  0, 9 . 3, 4 47 We have f 1 3, 4  0, 16 . Point at the equation f x  x 2 and then click on the button 2. in your computing toolbar. Then work out the expressions in parts (a) and (b) of the preceding exercise by pointing at them and clicking on the evaluate button. Supply each of the definitions f x  x 2 and g x  2 3. 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 Compute 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? 5. Given that f x  x 2 for all x a. b. c. 6. R and g x  1  x for all x R, simplify the following expressions: f g 0, 1 We have f g 0, 1  f g 0, 1  f 1, 2  1, 4 . g f 0, 1 We have g f 0, 1  g f 0, 1  g 0, 1  1, 2 . g g 0, 1 We have g g 0, 1  g g 0, 1  g 1, 2  2, 3 . a. Given that f x  3x 2 / x  1 for all x R 1 , determine whether or not f is one-one and find its range. Given any numbers x and y, the equation y  f x says that y  3x 2 x1 which gives us xy  y  3x 2 and finally 2y . x 3y Since, for each number y, there is only one number x that makes the equation y  3xx12 hold, the function f is one-one. Given any number y, the existence of a number x that makes the equation y  3xx12 hold will be 48 assured as long as y b. 3 and so the range of the function f is R 3. Point at the equation y  3x 2 x1 and ask Scientific Notebook to solve for x. How many values of x are given? Is this result consistent with the answer you gave in part a of the question? 7. Suppose that f : A B? Since B and that E 1 A. Is it true that E  f 1 f E ? What if f is one-one? What if f is onto f 1fE  x A fx fE f E . However, the equation E  f 1 f E we see at once that E f if f is onto the set B. For example, if f x  x 2 for 3 f 1 f 0, 1 x 3 then f is a function from x 3, 3 x2 0, 1  does not have to hold, even 3, 3 onto 0, 9 and 1, 1 0, 1 . On the other hand, if f is one-one then the equation E  f 1 f E must hold. Suppose that f is a one-one function from A to B and that E A. To see...
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