Download
1 / 17
170 likes | 293 Vues
This lecture focuses on the principles of set theory, including fundamental operations, laws, and methods of proof in discrete mathematics. Students will explore symmetric differences, unions, intersections, and identity laws, complemented by practical examples to solidify understanding. Emphasis is placed on comprehension rather than memorization of laws. The material includes insights into various proving techniques such as membership tables and logical equivalences. Assigned readings from Rosen provide further context for these concepts, ensuring students grasp the foundational elements of discrete structures.
E N D
Lecture 2.2: Set Theory* CS 250, Discrete Structures, Fall 2011 Nitesh Saxena *Adopted from previous lectures by Cinda Heeren
Course Admin • Slides from previous lectures all posted • HW1 Posted • Due at 11am 09/09/11 • Please follow all instructions • Recall: late submissions will not be accepted • Word Equation editor; Open Office; Alt-Codes • Please pick up your competency exams, if you haven’t done so Lecture 2.2 -- Set Theory
Outline • Set Theory, Operations and Laws Lecture 2.2 -- Set Theory
U A B like “exclusive or” Set Theory - Operators The symmetric difference, A B, is: A B = { x : (x A x B) v (x B x A)} = (A - B) U (B - A) Lecture 2.2 -- Set Theory
Set Theory - Operators A B = { x : (x A x B) v (x B x A)} = (A - B) U (B - A) Proof: { x : (x A x B) v (x B x A)} = { x : (x A - B) v (x B - A)} = { x : x ((A - B) U (B - A))} = (A - B) U (B - A) Lecture 2.2 -- Set Theory
Don’t memorize them, understand them! They’re in Rosen, p. 130 Set Theory - Famous Laws • Two pages of (almost) obvious. • One page of HS algebra. • One page of new. Lecture 2.2 -- Set Theory
A U = A A U U = U A U A = A A U = A A = A A = A Set Theory - Famous Laws • Identity • Domination • Idempotent Lecture 2.2 -- Set Theory
A U A = U A = A A A= Set Theory - Famous Laws • Excluded Middle • Uniqueness • Double complement Lecture 2.2 -- Set Theory
B U A B A A U (B U C) A U (B C) = A (B C) A (B U C) = Set Theory – Famous Laws • Commutativity • Associativity • Distributivity A U B = A B = (A U B)U C = (A B) C = (A U B) (A U C) (A B) U (A C) Lecture 2.2 -- Set Theory
(A UB)= A B (A B)= A U B Venn Diagrams are good for intuition, but we aim for a more formal proof. Set Theory – Famous Laws • DeMorgan’s I • DeMorgan’s II p q Lecture 2.2 -- Set Theory
New & important Like truth tables Not hard, a little tedious 3 Ways to prove Laws or set equalities • Show that A B and that A B. • Use a membership table. • Use logical equivalences to prove equivalent set definitions. Lecture 2.2 -- Set Theory
Example – the first way Prove that • () (x A U B) (x A U B) (x A and x B) (x A B) 2. () (x A B) (x A and x B) (x A U B) (x A U B) (A UB)= A B Lecture 2.2 -- Set Theory
(A UB)= A B Example – the second way Prove that using a membership table. 0 : x is not in the specified set 1 : otherwise Lecture 2.2 -- Set Theory
(A UB)= A B (A UB)= {x : (x A v x B)} = A B = {x : (x A) (x B)} Example – the third way Prove that using logically equivalent set definitions. = {x : (x A) (x B)} Lecture 2.2 -- Set Theory
Another example: applying the laws X (Y - Z) = (X Y) - (X Z). True or False? Prove your response. (X Y) - (X Z) = (X Y) (X Z)’ = (X Y) (X’ U Z’) = (X Y X’) U (X Y Z’) = U (X Y Z’) = (X Y Z’) Lecture 2.2 -- Set Theory
A U B = A = B A B = A-B = B-A = A Proof (direct and indirect) A B = Pv that if (A - B) U (B - A) = (A U B) then Suppose to the contrary, that A B , and that x A B. Then x cannot be in A-B and x cannot be in B-A. Then x is not in (A - B) U (B - A). But x is in A U B since (A B) (A U B). Thus, A B = . Lecture 2.2 -- Set Theory
Today’s Reading • Rosen 2.1 and 2.2 Lecture 2.2 -- Set Theory
