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CS 173: Discrete Mathematical Structures

Cs173 - Spring 2004. CS 173 Announcements. Homework 2 returned in section this week.Homework 3 available. Due 09/17, 8a.. . . Cs173 - Spring 2004. CS 173 Set Theory - Definitions and notation. A set is an unordered collection of elements.Some examples:{1, 2, 3} is the set containing

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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 2 returned in section this week. Homework 3 available. Due 09/17, 8a.

    3. Cs173 - Spring 2004 CS 173 Set Theory - Definitions and notation A set is an unordered collection of elements. Some examples: {1, 2, 3} is the set containing “1” and “2” and “3.” {1, 1, 2, 3, 3} = {1, 2, 3} since repetition is irrelevant. {1, 2, 3} = {3, 2, 1} since sets are unordered. {1, 2, 3, …} is one way we denote an infinite set (in this case, the natural numbers). ? = {} is the empty set, or the set containing no elements.

    4. Cs173 - Spring 2004 CS 173 Set Theory - Definitions and notation x ? S means “x is an element of set S.” x ? S means “x is not an element of set S.” A ? B means “A is a subset of B.”

    5. Cs173 - Spring 2004 CS 173 Set Theory - Definitions and notation A ? B means “A is a subset of B.” A ? B means “A is a superset of B.” A = B if and only if A and B have exactly the same elements.

    6. Cs173 - Spring 2004 CS 173 Set Theory - Definitions and notation A ? B means “A is a proper subset of B.” A ? B, and A ? B. ?x ((x ? A) ? (x ? B)) ? ??x ((x ? B) ? (x ? A)) ?x ((x ? A) ? (x ? B)) ? ?x ?(?(x ? B) v (x ? A)) ?x ((x ? A) ? (x ? B)) ? ?x ((x ? B) ? ?(x ? A)) ?x ((x ? A) ? (x ? B)) ? ?x ((x ? B) ? (x ? A))

    7. Cs173 - Spring 2004 CS 173 Set Theory - Definitions and notation Quick examples: {1,2,3} ? {1,2,3,4,5} {1,2,3} ? {1,2,3,4,5} Is ? ? {1,2,3}?

    8. Cs173 - Spring 2004 CS 173 Set Theory - Definitions and notation Quiz time: Is {x} ? {x}?

    9. Cs173 - Spring 2004 CS 173 Set Theory - Ways to define sets Explicitly: {John, Paul, George, Ringo} Implicitly: {1,2,3,…}, or {2,3,5,7,11,13,17,…} Set builder: { x : x is prime }, { x | x is odd }. In general { x : P(x) is true }, where P(x) is some description of the set.

    10. Cs173 - Spring 2004 CS 173 Set Theory - Cardinality If S is finite, then the cardinality of S, |S|, is the number of distinct elements in S.

    11. Cs173 - Spring 2004 CS 173 Set Theory - Power sets If S is a set, then the power set of S is 2S = { x : x ? S }.

    12. Cs173 - Spring 2004 CS 173 Set Theory - Cartesian Product The Cartesian Product of two sets A and B is: A x B = { <a,b> : a ? A ? b ? B}

    13. Cs173 - Spring 2004 CS 173 Set Theory - Operators The union of two sets A and B is: A ? B = { x : x ? A v x ? B}

    14. Cs173 - Spring 2004 CS 173 Set Theory - Operators The intersection of two sets A and B is: A ? B = { x : x ? A ? x ? B}

    15. Cs173 - Spring 2004 CS 173 Set Theory - Operators The intersection of two sets A and B is: A ? B = { x : x ? A ? x ? B}

    16. Cs173 - Spring 2004 CS 173 Set Theory - Operators The intersection of two sets A and B is: A ? B = { x : x ? A ? x ? B}

    17. Cs173 - Spring 2004 CS 173 Set Theory - Operators The complement of a set A is: A = { x : x ? A}

    18. Cs173 - Spring 2004 CS 173 Set Theory - Operators The set difference, A - B, is:

    19. Cs173 - Spring 2004 CS 173 Set Theory - Operators The symmetric difference, A ? B, is: A ? B = { x : (x ? A ? x ? B) v (x ? B ? x ? A)}

    20. Cs173 - Spring 2004 CS 173 Set Theory - Operators A ? B = { x : (x ? A ? x ? B) v (x ? B ? x ? A)}

    21. Cs173 - Spring 2004 CS 173 Set Theory - Famous Identities Two pages of (almost) obvious. One page of HS algebra. One page of new.

    22. Cs173 - Spring 2004 CS 173 Set Theory - Famous Identities Identity Domination Idempotent

    23. Cs173 - Spring 2004 CS 173 Set Theory - Famous Identities Excluded Middle Uniqueness Double complement

    24. Cs173 - Spring 2004 CS 173 Set Theory - Famous Identities Commutativity Associativity Distributivity

    25. Cs173 - Spring 2004 CS 173 Set Theory - Famous Identities DeMorgan’s I DeMorgan’s II

    26. Cs173 - Spring 2004 CS 173 Set Theory - 4 Ways to prove identities Show that A ? B and that A ? B. Use a membership table. Use previously proven identities. Use logical equivalences to prove equivalent set definitions.

    27. Cs173 - Spring 2004 CS 173 Set Theory - 4 Ways to prove identities Prove that (?) 2. (?)

    28. Cs173 - Spring 2004 CS 173 Set Theory - 4 Ways to prove identities Prove that using a membership table. 0 : x is not in the specified set 1 : otherwise

    29. Cs173 - Spring 2004 CS 173 Set Theory - 4 Ways to prove identities Prove that using identities.

    30. Cs173 - Spring 2004 CS 173 Set Theory - 4 Ways to prove identities Prove that using logically equivalent set definitions.

    31. Cs173 - Spring 2004 CS 173 Set Theory - A proof for us to do together. X ? (Y - Z) = (X ? Y) - (X ? Z). True or False? Prove your response.

    32. Cs173 - Spring 2004 CS 173 Set Theory - A proof for us to do together. Pv that if (A - B) U (B - A) = (A U B) then ______

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