Set Theory Basics of Set Theory. mjarrar Watch this lecture and download the slides
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1 9/6/17 Birzeit University Palestine Basics of 6.2 Properties of Sets and Element Argument 6.3 Algebraic Proofs mjarrar Watch this lecture and download the slides Course Page: More Online Courses at: Acknowledgement: This lecture is based on (but not limited to) to chapter 5 in Discrete Mathematics with Applications by Susanna S. Epp (3rd Edition). 2 1
2 Birzeit University Palestine 2015 In this lecture: qpart 1: History of Algebra qpart 2: What is Boolean Algebra qpart 3: Proving Boolean Algebra Properties 3 What is Algebra? Al-Khwarizmi (Baghdad) Developed an advanced arithmetical system with which they were able to do calculations in an algorithmic fashion. The Compendious Book on Calculation by Completion and Balancing Statements to describe relationships between things Symbols and the rules for manipulating these symbols )? جبر) Do you know any algebra 4 2
3 Birzeit University Palestine 2015 In this lecture: qpart 1: History of Algebra qpart 2: What is Boolean Algebra qpart 3: Proving Boolean Algebra Properties 5 Boolean Algebra Introduced by George Boole in his first book The Mathematical Analysis of Logic (1847) A structure abstracting the computation with the truth values false and true. George Boole England Instead of elementary algebra where the values of the variables are numbers and the main operations are addition and multiplication the main operations of Boolean algebra are the conjunction ( ) the disjunction ( ) and the negation not ( ). Used extensively in the simplification of logic Circuits 6 3
4 Compare Logical Equivalences For all statement variables p qandr: Set Properties For all sets A B andc: a. p q q p a. A B = B A b. p q q p b. A B = B A a. p (q r) p (q r) a. A (B C) A (B C) b. p (q r) p (q r) b. A (B C) A (B C) a. p (q r) (p q) (p r) a. A (B C) (A B) (A C) b. p (q r) (p q) (p r) b. A (B C) (A B) (A C) a. p c p a. A = A b. p t p b. A U = A a. p p t a. A A c = U b. p p c b. A A c = ( p) p Both are special cases of the same general (A c ) c = A structure known as a Boolean Algebra. a. p p p a. A A = A b. p p p b. A A = A a. p t t a. A U = U b. c c b. = 7 Boolean Algebra Definition: Boolean Algebra A Boolean algebra is a set B together with two operations generally denoted + and suchthatforalla and b in B both a + b and a b are in B and the following properties hold: 1. Commutative Laws: For all a and b in B (a) a + b = b + a and (b) a b = b a. 2. Associative Laws: For all a b andc in B (a) (a + b) + c = a + (b + c) and (b) (a b) c = a (b c). 3. Distributive Laws: For all a b and c in B (a) a + (b c) = (a + b) (a + c) and (b) a (b + c) = (a b) + (a c). 4. Identity Laws: There exist distinct elements 0 and 1 in B such that for all a in B (a) a + 0 = a and (b) a 1 = a. 5. Complement Laws: For each a in B there exists an element in B denoted a and called the complement or negation of a suchthat (a) a + a = 1 and (b)a a =
5 Properties of a Boolean Algebra Theorem Properties of a Boolean Algebra Let B be any Boolean algebra. 1. Uniqueness of the Complement Law: For all a and x in B ifa + x = 1and a x = 0thenx = a. 2. Uniqueness of 0 and 1: If there exists x in B such that a + x = a for all a in B then x = 0 and if there exists y in B such that a y = a for all a in Btheny = Double Complement Law: For all a B (a) = a. 4. Idempotent Law: For all a B 5. Universal Bound Law: For all a B (a) a + a = a and (b) a a = a. (a) a + 1 = 1 and (b) a 0 = De Morgan s Laws: For all a and b B 7. Absorption Laws: For all a and b B 8. Complements of 0 and 1: (a) a + b = a b and (b) a b = a + b. (a) (a + b) a = a and (b) (a b) + a = a. (a) 0 = 1 and (b)1 = 0. continued on page Birzeit University Palestine 2015 In this lecture: qpart 1: History of Algebra qpart 2: What is Boolean Algebra qpart 3: Proving Boolean Algebra Properties 10 5
6 Proving of Boolean Algebra Properties Proof: Suppose a and x are particular but arbitrarily chosen elements of B that satisfy the following hypothesis: a + x = 1 and a x = 0. Then 11 Proving of Boolean Algebra Properties Suppose B is a Boolean algebra and a is any element of B. Then and 12 6
Set Theory Basics of Set Theory. 6.2 Properties of Sets and Element Argument. 6.3 Algebraic Proofs 6.4 Boolean Algebras.
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