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CS 173: Discrete Mathematical Structures

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CS 173: Discrete Mathematical Structures

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    1. CS 173: Discrete Mathematical Structures Cinda Heeren heeren@cs.uiuc.edu Rm 2213 Siebel Center Office Hours: M 11a-12p

    2. Cs173 - Spring 2004 CS 173 Announcements Homework 3 returned in section this week. Homework 4 available. Due 09/24, 8a.

    3. Cs173 - Spring 2004 CS 173 Functions - misc. properties f-1(?) = ? f-1(A U B) = f-1(A) U f-1(B) f-1(A ? B) = f-1(A) ? f-1(B)

    4. Cs173 - Spring 2004 CS 173 Functions - injection A function f: A ? B is one-to-one (injective, an injection) if ?a,b,c, (f(a) = b ? f(c) = b) ? a = c

    5. Cs173 - Spring 2004 CS 173 Functions - surjection A function f: A ? B is onto (surjective, a surjection) if ?b ? B, ?a ? A f(a) = b

    6. Cs173 - Spring 2004 CS 173 Functions - bijection A function f: A ? B is bijective if it is one-to-one and onto.

    7. Cs173 - Spring 2004 CS 173 Functions - examples Suppose f: R+ ? R+, f(x) = x2. Is f one-to-one? Is f onto? Is f bijective?

    8. Cs173 - Spring 2004 CS 173 Functions - examples Suppose f: R ? R+, f(x) = x2. Is f one-to-one? Is f onto? Is f bijective?

    9. Cs173 - Spring 2004 CS 173 Functions - examples Suppose f: R ? R, f(x) = x2. Is f one-to-one? Is f onto? Is f bijective?

    10. Cs173 - Spring 2004 CS 173 Functions - composition Let f:A?B, and g:B?C be functions. Then the composition of f and g is: (g o f)(x) = g(f(x))

    11. Cs173 - Spring 2004 CS 173 Functions - a little problem Let f:A?B, and g:B?C be functions. Prove that if f and g are one to one, then g o f :A?C is one to one.

    12. Cs173 - Spring 2004 CS 173 Functions - another Let f:A?B, and g:B?C be functions. Prove that if f and g are onto, then g o f :A?C is onto.

    13. Cs173 - Spring 2004 CS 173 Familiar functions Polynomials: f(x) = a0xn + a1xn-1 + … + an-1x1 + anx0 Ex: f(x) = x3 - 2x2 + 15 Exponentials: f(x) = cdx Ex: f(x) = 310x, f(x) = ex Logarithms: log2 x = y, where 2y = x.

    14. Cs173 - Spring 2004 CS 173 Familiar functions Ceiling: f(x) = ?x? the least integer y so that x ? y. Ex: ?1.2? = 2; ?-1.2? = -1; ?1? = 1 Floor: f(x) = ?x? the greatest integer y so that x ? y. Ex: ?1.8? = 1; ?-1.8? = -2; ?-5? = -5 Quiz: what is ?-1.2 + ?1.1?? ?

    15. Cs173 - Spring 2004 CS 173 Two example algorithms. Suppose we wish to find the maximum number in a sequence of n numbers. How long should we spend doing this?

    16. Cs173 - Spring 2004 CS 173 Two example algorithms. I have a number between 0 and 63. You ask a question, I’ll tell you yes or no. How long will it take you to find my secret number?

    17. Cs173 - Spring 2004 CS 173 Who wins the race? The following graph gives times for completing races of length x, for 4 different competitors.

    18. Cs173 - Spring 2004 CS 173 Quiz time… Describe these functions:

    19. Cs173 - Spring 2004 CS 173 Quiz time… Describe this function:

    20. Cs173 - Spring 2004 CS 173 Quiz time… Describe this function:

    21. Cs173 - Spring 2004 CS 173 Quiz time… Describe this function:

    22. Cs173 - Spring 2004 CS 173 Growth of functions Algorithm analysis is concerned with: Type of function that describes run time (we ignore constant factors since different machines have different speed/cycle) Large values of x

    23. Cs173 - Spring 2004 CS 173 Growth of functions Important definition: For functions f and g we write f(x) = O(g(x)) to denote ? c,k so that ? x>k, f(x) ? c·g(x)

    24. Cs173 - Spring 2004 CS 173 Growth of functions f(x) = O(g(x)) iff ? c,k so that ? x>k, f(x) ? c·g(x)

    25. Cs173 - Spring 2004 CS 173 Growth of functions (examples) f(x) = O(g(x)) iff ? c,k so that ? x>k, f(x) ? c·g(x)

    26. Cs173 - Spring 2004 CS 173 Growth of functions (examples) f(x) = O(g(x)) iff ? c,k so that ? x>k, f(x) ? c·g(x)

    27. Cs173 - Spring 2004 CS 173 Growth of functions (examples) f(x) = O(g(x)) iff ? c,k so that ? x>k, f(x) ? c·g(x)

    28. Cs173 - Spring 2004 CS 173 Growth of functions (examples) f(x) = O(g(x)) iff ? c,k so that ? x>k, f(x) ? c·g(x)

    29. Cs173 - Spring 2004 CS 173 Growth of functions (examples) f(x) = O(g(x)) iff ? c,k so that ? x>k, f(x) ? c·g(x)

    30. Cs173 - Spring 2004 CS 173 Growth of functions (examples)

    31. Cs173 - Spring 2004 CS173 Growth of functions Guidelines: In general, only the largest term in a sum matters. a0xn + a1xn-1 + … + an-1x1 + anx0 = O(xn) n dominates lg n. n5lg n = O(n6) List of common functions in increasing O() order: 1 n (n lg n) n2 n3 … 2n n!

    32. Cs173 - Spring 2004 CS173 Growth of functions (more examples) So

    33. Cs173 - Spring 2004 CS173 Growth of functions - a theorem If f1(x) = O(g1(x)) and f2(x)=O(g2(x)), then f1(x) + f2(x) = O(max{g1(x),g2(x)})

    34. Cs173 - Spring 2004 CS173 Growth of functions - a corollary We know: If f1(x) = O(g1(x)) and f2(x)=O(g2(x)), then f1(x) + f2(x) = O(max{g1(x),g2(x)})

    35. Cs173 - Spring 2004 CS173 Growth of functions - another theorem If f1(x) = O(g1(x)) and f2(x)=O(g2(x)), then f1(x)·f2(x) = O(g1(x)·g2(x))

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