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CSE 311 Foundations of Computing I

CSE 311 Foundations of Computing I. Lecture 10 Set Theory Autumn 2012. Announcements. Reading assignments Wednesday : 4.1-4.2 7 th Edition 3.4, 3.6 up to p. 227 6 th Edition 2.4, 2.5 up to p. 177 5 th Edition Homework 4 Coming soon. Set Theory.

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CSE 311 Foundations of Computing I

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  1. CSE 311 Foundations of Computing I Lecture 10 Set TheoryAutumn 2012 CSE 311

  2. Announcements • Reading assignments • Wednesday: • 4.1-4.2 7th Edition • 3.4, 3.6 up to p. 227 6th Edition • 2.4, 2.5 up to p. 177 5th Edition • Homework 4 • Coming soon . . . CSE 311

  3. Set Theory • Formal treatment dates from late 19th century • Direct ties between set theory and logic • Important foundational language CSE 311

  4. Definition: A set is an unordered collection of objects x A : “x is an element of A” “x is a member of A” “x is in A” x A :  (x A) / Give some examples: Finite sets, Multiple domains N, Z, Q, R Emptyset Sets containing sets CSE 311

  5. Definitions • A and B are equal if they have the same elements • A is a subset of B if every element of A is also in B A = B x (x A x B) A  B x (x A x B) A  B CSE 311

  6. Empty Set and Power Set • Empty set ∅ does not contain any elements • Power set of a set A = set of all subsets of A 𝓟(A) = { B : B  A} CSE 311

  7. Cartesian Product : A  B A  B = { (a, b) | a  A  b  B} CSE 311

  8. Set operations A  B = { x | (x A)  (x B) } union A  B = { x | (x A)  (x B) } intersection / A - B = { x | (x A)  (x  B) } set difference symmetric difference A  B = { x | (x A)  (x B) } _ A = { x | x A } (with respect to universe U) / complement CSE 311

  9. It’s Boolean algebra again • Definition for based on  • Definition for based on  • Complement works like  CSE 311

  10. De Morgan’s Laws A  B = A  B A  B = A  B A B Proof technique: To show C = D show x C x  D and x D x C Prove A  B = A  B Begin with x  A  B  x  A  x  B CSE 311

  11. Distributive Laws A  (B  C) = (A  B)  (A  C) A  (B  C) = (A  B)  (A  C) A B A B C C CSE 311

  12. Characteristic vectors:Representing sets using bits • Suppose universe U is {1,2,...,n} • Can represent set B ⊆ U as a vector of bits: b1b2...bn where bi=1  (i ∈B) bi=0  (i ∈B) • Called the characteristic vector of set B • Given characteristic vectors for A and B • What is characteristic vector for A  B? A  B ? / CSE 311

  13. Boolean operations on bit-vectors:(a.k.a. bit-wise operations) • 01101101 Java: z=x|y 00110111 01111111 • 00101010         Java: z=x&y00001111 00001010 • 01101101 Java: z=x^y00110111 01011010 CSE 311

  14. A simple identity • If x and y are bits: (x y)  y = ? • What if x and y are bit-vectors? CSE 311

  15. Private Key Cryptography • Alice wants to be able to communicate message secretly to Bob so that eavesdropper Eve who hears their conversation, cannot tell what Alice’s message is. • Alice and Bob can get together and privately share a secret key K ahead of time. CSE 311

  16. One-time pad • Alice and Bob privately share random n-bit vector K • Eve does not know K • Later, Alice has n-bit message m to send to Bob • Alice computes C = m  K • Alice sends C to Bob • Bob computes m = C  K which is (m  K)  K • Eve cannot figure out m from C unless she can guess K CSE 311

  17. Unix/Linux file permissions • ls–l drwxr-xr-x ... Documents/ -rw-r--r-- ... file1 • Permissions maintained as bit vectors • Letter means bit is 1 -means bit is 0. CSE 311

  18. Russell’s Paradox S = { x | xx } / CSE 311

  19. Functions review • A function from A to B • an assignment of exactly one element of B to each element of A. • We write f: A→B. • “Image of a” = f(a) • Domain of f : A • Range of f = set of all images of elements of A CSE 311

  20. Image, Preimage A B 1 a b 2 c 3 d 4 e CSE 311

  21. Is this a function? one-to-one?onto? 1 a b 2 c 3 d 4 e 5 6 CSE 311

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