1 / 21

Set Theory Foundations in Computing: Lecture 10 Overview

This lecture focuses on the foundations of set theory as it relates to computing, exploring key concepts like the definition of sets, equality, subsets, and operations including union, intersection, and differences. It highlights the empty set and power set, Cartesian products, and Boolean algebra in set theory. Students will also learn about characteristic vectors for representation of sets and the implications of set operations on bit vectors. Furthermore, the lecture touches on cryptographic key sharing and the importance of functions in computing.

howard
Download Presentation

Set Theory Foundations in Computing: Lecture 10 Overview

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related