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CSE 311: Foundations of Computing

CSE 311: Foundations of Computing. Fall 2013 Lecture 9: Set theory and functions. announcements. Reading assignment Set theory 2.1-2.3 (both editions). set theory. Formal treatment dates from late 19 th century Direct ties between set theory and logic Important foundational language.

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CSE 311: Foundations of Computing

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  1. CSE 311: Foundations of Computing Fall 2013 Lecture 9: Set theory and functions

  2. announcements Reading assignment Set theory 2.1-2.3 (both editions)

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

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

  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

  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

  7. cartesian product

  8. set operations union intersection set difference symmetric difference (with respect to universe U) complement

  9. it’s Boolean algebra again • Definition for  based on  • Definition for  based on  • Complement works like 

  10. De Morgan’s Laws 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

  11. distributive laws A B A B C C

  12. representing sets using bits • Suppose universe is • Can represent set as a vector of bits: wherewhen when • Called the characteristic vector of set B • Given characteristic vectors for and • What is characteristic vector for ? ?

  13. bitwise operations on vectors 01101101 Java: z=x|y00110111 01111111 00101010         Java: z=x&y  00001111 00001010 01101101 Java: z=x^y  00110111 01011010

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

  15. private key cryptography • Alice wants 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.

  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

  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.

  18. russell’s paradox

  19. functions review • A function from to • an assignment of exactly one element of to each element of . • We write . • “Image of a” = • Domain of: • Range of = set of all images of elements of

  20. image, preimage 1 a b 2 c 3 d 4 e

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

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