Содержание
- 2. Copyright © Cengage Learning. All rights reserved. Functions Defined on General Sets SECTION 7.1
- 3. Functions Defined on General Sets We have already defined a function as a certain type of
- 4. Arrow Diagrams
- 5. Arrow Diagrams We have known that if X and Y are finite sets, you can define
- 6. Arrow Diagrams This arrow diagram does define a function because 1. Every element of X has
- 7. Example 2 – A Function Defined by an Arrow Diagram Let X = {a, b, c}
- 8. Example 2 – Solution a. domain of co-domain of b. c. range of d. Yes, No
- 9. Arrow Diagrams In Example 2 there are no arrows pointing to the 1 or the 3.
- 10. Arrow Diagrams Earlier we have given a test for determining whether two functions with the same
- 11. Example 3 – Equality of Functions a. Let J3 = {0, 1, 2}, and define functions
- 12. Example 3 – Solution a. Yes, the table of values shows that f (x) = g(x)
- 13. Examples of Functions
- 14. Example 4 – The Identity Function on a Set Given a set X, define a function
- 15. Example 4 – Solution Whatever is input to the identity function comes out unchanged, so and
- 16. Examples of Functions
- 17. Example 8 – The Logarithmic Function with Base b Find the following: a. b. c. d.
- 18. Example 8 – Solution d. because the exponent to which 2 must be raised to obtain
- 19. Examples of Functions We have known that if S is a nonempty, finite set of characters,
- 20. Example 9 – Encoding and Decoding Functions Digital messages consist of finite sequences of 0’s and
- 21. Example 9 – Encoding and Decoding Functions The receiver of the message decodes it by replacing
- 22. Example 9 – Encoding and Decoding Functions The encoding function E is the function from A
- 23. Example 9 – Encoding and Decoding Functions The advantage of this particular coding scheme is that
- 24. Boolean Functions
- 25. Boolean Functions We have discussed earlier that how to find input/output tables for certain digital logic
- 26. Boolean Functions
- 27. Example 11 – A Boolean Function Consider the three-place Boolean function defined from the set of
- 28. Example 11 – Solution The rest of the values of f can be calculated similarly to
- 29. Checking Whether a Function Is Well Defined
- 30. Checking Whether a Function Is Well Defined It can sometimes happen that what appears to be
- 31. Checking Whether a Function Is Well Defined For instance, when x = 2, there is no
- 32. Example 12 – A Function That Is Not Well Defined We know that Q represents the
- 33. Example 12 – Solution The function f is not well defined. The reason is that fractions
- 34. Example 12 – Solution But applying the formula for f, you find that and so This
- 35. Checking Whether a Function Is Well Defined Note that the phrase well-defined function is actually redundant;
- 36. Functions Acting on Sets
- 37. Functions Acting on Sets Given a function from a set X to a set Y, you
- 38. Example 13 – The Action of a Function on Subsets of a Set Let X =
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