CS1022 Computer Programming & Principles Lecture 9.1 Boolean Algebra (1)

Plan of lecture Introduction Boolean operations Boolean algebra vs. logic Boolean expressions Laws of Boolean algebra Boolean functions Disjunctive normal form CS1022

Introduction Boolean algebra Is a logical calculus Structures and rules applied to logical symbols Just like ordinary algebra is applied to numbers Uses a two-valued set {0, 1} Operations: conjunction, disjunction and negation Is related with Propositional logic and Algebra of sets Underpins logical circuits (electronic components) CS1022

Boolean operations (1)  1  1 1 0 1 Boolean algebra uses set B  0, 1 of values Operations: Disjunction (), Conjunction () and Negation ()  1  1 1 0 1 CS1022

Boolean operations (2) p q p  q p  q 1 p p 1 If p and q are Boolean variables “Variables” because they vary over set B  0, 1 That is, p, q  B We can build the following tables: p q p  q p  q 1 p p 1 CS1022

Boolean algebra vs. logic Boolean algebra is similar to propositional logic Boolean variables p, q,... like propositions P, Q,... Values B  0, 1 like F (false) and T (true) Operators are similar: p  q similar to P or Q p  q similar to P and Q p similar to not P Tables also called “truth tables” (as in logic) CS1022

Boolean expressions As in logic, we can build more complex constructs from basic variables and operators ,  and  Complex constructs are called Boolean expressions A recursive definition for Boolean expressions: All Boolean variables (p, q, r, s, etc.) are Boolean exprns. If α and β are Boolean expressions then α is a Boolean expression (and so is β) (α  β) is a Boolean expression (α  β) is a Boolean expression Example: ((p  q)  (r  s)) CS1022

Equivalence of expressions Two Boolean expressions are equivalent if they have the same truth table The final outcome of the expressions are the same, for any value the variables have We can prove mechanically if any two expressions are equivalent or not “Mechanic” means “a machine (computer) can do it” No need to be creative, clever, etc. An algorithm explains how to prove equivalence CS1022

Laws of Boolean algebra (1) Some equivalences are very useful They simplify expressions, making life easier They allow us to manipulate expressions Some equivalences are “laws of Boolean algebra” We have, in addition to truth-tables, means to operate on expressions Because equivalence is guaranteed we won’t need to worry about changing the meaning CS1022

Laws of Boolean algebra (2) Commutative Laws p  q  q  p p  q  q  p Associativity Laws p  (q  r)  (p  q)  r p  (q  r)  (p  q)  r Distributive Laws p  (q  r)  (p  q)  (p  r) p  (q  r)  (p  q)  (p  r) Idempotent Laws p  p  p p  p  p Absorption Laws p  (p  q)  p p  (p  q)  p De Morgan’s Laws (p  q)  p  q (p  q)  p  q CS1022

Laws of Boolean algebra (3) Do they look familiar? Logics, sets & Boolean algebra are all related Summary of correspondence: Logical operation Set operation Boolean operation not   or   and   CS1022

Laws of Boolean algebra (4) Very important: Laws shown with Boolean variables (p, q, r, etc.) However, each Boolean variable stand for an expression Wherever you see p, q, r, etc., you should think of “place holders” for any complex expression A more “correct” way to represent e.g. idempotent law:      CS1022

Laws of Boolean algebra (5) What do we mean by “place holders”? The laws are “templates” which can be used in many different situations For example: ((p  q)  (q  s))  ((p  q)  (q  s))     ((p  q)  (q  s)) CS1022

Laws of Boolean algebra (6) We should be able to prove all laws We know how to build truth tables (lectures 2.1 & 2.2) Could you work out an algorithm to do this? There is help at hand... CS1022

Laws of Boolean algebra (7) Using the laws of Boolean algebra, show that (p  q)  (p  q) is equivalent to p Solution: (p  q)  (p  q)  (p  q)  (p  q) De Morgan’s  (p  q)  (p  q) since p  p  p  (q  q) Distr. Law  p  0 since (q q)  0  p from def. of  CS1022

Boolean functions (1) A Boolean function of n variables p1, p2, ..., pn is A function f : Bn  B Such that f(p1, p2, ..., pn) is a Boolean expression Any Boolean expression can be represented in an equivalent standard format The so-called disjunctive normal form Format to make it easier to “process” expressions Simpler, fewer operators, etc. CS1022

Boolean functions (2) Example: function minterm m(p, q, r) It has precisely one “1” in the final column of truth table m(p, q, r)  1 if p  0, q  1 and r  1 We can define m(p, q, r)  p q  r The expression p q  r is the product representation of the minterm m p q r m 1 CS1022

Boolean functions (3) p q r m 1 Any minterm m(p1, p2, ..., pr) can be represented as a conjunction of the variables pi or their negations Here’s how: Look at row where m takes value 1 If pi  1, then pi is in the product representation of m If pi  0, then pi is in the product representation of m We thus have m(p, q, r)  p q  r p q r m 1 CS1022

Disjunctive normal form (1) We can express any Boolean function uniquely as a disjunction (a sequence of “”) of minterms This is the disjunctive normal form The idea is simple: Each minterm is a conjunction (sequence of “”) which, if all holds, then we have output (result) “1” The disjunction of minterms lists all cases which, if at least one holds, then we have output (result) “1” In other words: we list all cases when it should be “1” and say “at least one of these is 1” CS1022

Disjunctive normal form (2) Find disjunctive normal form for (p  q)  (q r) Let f  (p  q)  (q r). Its truth table is as below The minterms are (rows with f  1) p  q r p  q r p  q  r p  q  r Disjunctive normal form is (p  q r)  (p  q r)  (p  q  r)  (p  q  r) p q r f 1 p q r f 1 p q r f 1 p q r f 1 p q r f 1 p q r f 1 CS1022

Minimal set of operators Any Boolean function can be represented with Boolean operators , ,  This is a complete set of operators “Complete” = “we can represent anything we need” It is not minimal, though: De Morgan’s (p  q)  p  q, so (p  q)  (p  q) We can define “” in terms of ,  Minimal set of Boolean operators: ,  or ,  Actually, any two operators suffice Minimality comes at a price: expressions become very complex and convoluted CS1022

Further reading R. Haggarty. “Discrete Mathematics for Computing”. Pearson Education Ltd. 2002. (Chapter 9) Wikipedia’s entry on Boolean algebra Wikibooks entry on Boolean algebra CS1022