# University of Aberdeen, Computing Science CS2013 Mathematics for Computing Science Adam Wyner Slides adapted from Michael P. Frank ’ s course based on.

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University of Aberdeen, Computing Science CS2013 Mathematics for Computing Science Adam Wyner Slides adapted from Michael P. Frank ’ s course based on the text Discrete Mathematics & Its Applications (5 th Edition) by Kenneth H. Rosen

Propositional Logic Rosen 5 th ed., §§1.1-1.2 (but much extended) ~85 slides, ~2 lectures

Foundations of Logic Mathematical Logic is a tool for working with compound statements. It includes: A formal language for expressing them. A methodology for reasoning about their truth or falsity. It is the foundation for expressing formal proofs in all branches of mathematics. Fall 2013Frank / van Deemter / Wyner2

Two Logical Systems: 1.Propositional logic 2.Predicate logic (extends 1. ) Many other logical `calculi` exist, but they tend to resemble these two Many other logical `calculi` exist, but they tend to resemble these two Fall 2013Frank / van Deemter / Wyner3

Propositional Logic (§1.1) Propositional Logic is the logic of compound statements built from simpler statements using so-called Boolean connectives. Some applications in computer science: Design of digital electronic circuits.Design of digital electronic circuits. Expressing conditions in programs.Expressing conditions in programs. Queries to databases & search engines.Queries to databases & search engines. George Boole (1815-1864) Chrysippus of Soli (ca. 281 B.C. – 205 B.C.) Fall 2013Frank / van Deemter / Wyner4

Propositions in natural language In propositional logic, a proposition is simply: a statement (i.e., a declarative sentence)a statement (i.e., a declarative sentence) –with some definite meaning having a truth value that ’ s either true (T) or false (F). Only values statements can have.having a truth value that ’ s either true (T) or false (F). Only values statements can have. –Never both, or somewhere “ in between ”. However, you might not know the truth value Fall 2013Frank / van Deemter / Wyner5

Examples of NL Propositions “ It is raining. ” (In a given situation.)“ It is raining. ” (In a given situation.) “ Beijing is the capital of China, and 1 + 2 = 2 ”“ Beijing is the capital of China, and 1 + 2 = 2 ” But, the following are NOT propositions: “ Who ’ s there? ” (interrogative: no truth value)“ Who ’ s there? ” (interrogative: no truth value) “ x := x+1 ” (imperative: no truth value)“ x := x+1 ” (imperative: no truth value) “ 1 + 2 ” (term: no truth value)“ 1 + 2 ” (term: no truth value) Fall 2013Frank / van Deemter / Wyner6

Propositions in Propositional Logic Atoms: p, q, r, … (Corresponds with simple English sentences, e.g. ‘ I had salad for lunch ’ )Atoms: p, q, r, … (Corresponds with simple English sentences, e.g. ‘ I had salad for lunch ’ ) Complex propositions : built up from atoms using operators: p  q (Corresponds with compound English sentences, e.g., “I had salad for lunch and I had steak for dinner. ” )Complex propositions : built up from atoms using operators: p  q (Corresponds with compound English sentences, e.g., “I had salad for lunch and I had steak for dinner. ” ) Fall 2013Frank / van Deemter / Wyner7

Defining Propositions Logic defines notions of atomic and complex propositions and what complex propositions “mean”.Logic defines notions of atomic and complex propositions and what complex propositions “mean”. We explain by example, giving precise definitions.We explain by example, giving precise definitions. Fall 2013Frank / van Deemter / Wyner8

An operator or connective combines with n operand expressions into a larger expression. Unary operators take 1 operand;Unary operators take 1 operand; Binary operators take 2 operands.Binary operators take 2 operands. Propositional or Boolean operators operate on propositions instead of on numbers (+,-).Propositional or Boolean operators operate on propositions instead of on numbers (+,-). Operators / Connectives Fall 2013Frank / van Deemter / Wyner9

Common Boolean Operators Formal Name NicknameAritySymbol Negation operator NOTUnary¬ Conjunction operator ANDBinary Disjunction operator ORBinary Exclusive-OR operator XORBinary Implication operator IMPLIESBinary Biconditional operator IFFBinary↔ Fall 2013Frank / van Deemter / Wyner10

The Negation Operator The unary negation operator “ ¬ ” (NOT) combines with one prop, transforming the prop into its negation. E.g. If p = “ I have brown hair. ” then ¬p = “ I do not have brown hair. ” then ¬p = “ I do not have brown hair. ” The truth table for NOT: T :≡ True; F :≡ False “ :≡ ” means “ is defined as ” Operand column Result column Fall 2013Frank / van Deemter / Wyner11

Truth-functionality Truth table expresses truth/falsity of ¬p in terms of truth/falsity of pTruth table expresses truth/falsity of ¬p in terms of truth/falsity of p This not possible for the operator ‘tomorrow’, or `probably’:This not possible for the operator ‘tomorrow’, or `probably’: –‘Tomorrow p’ is true iff p is ….’?? –‘Probably p’ is true iff p is ….’?? Fall 2013Frank / van Deemter / Wyner12

Truth-functionality Truth table expresses truth/falsity of ¬p in terms of truth/falsity of p.Truth table expresses truth/falsity of ¬p in terms of truth/falsity of p. Each horizontal line of the table expresses some alternative context.Each horizontal line of the table expresses some alternative context. Truth-functional operator: an operator that is a function from the truth values of the component expressions to a truth value.Truth-functional operator: an operator that is a function from the truth values of the component expressions to a truth value. NOT is truth functional. Yesterday is not.NOT is truth functional. Yesterday is not. Propositional logic is only about truth-functional operators.Propositional logic is only about truth-functional operators. We can compute the values of the complex expressions.We can compute the values of the complex expressions. Fall 2013Frank / van Deemter / Wyner13

Comment on Truth Not concerned with the "meaning" of a proposition other than whether it is true or false. Not about "how" we know it is true or false, but supposing it is, what else do we know. Abstraction.Not concerned with the "meaning" of a proposition other than whether it is true or false. Not about "how" we know it is true or false, but supposing it is, what else do we know. Abstraction. The "truth" of a proposition determined "by inspection" – The book is on the table. The "real" world.The "truth" of a proposition determined "by inspection" – The book is on the table. The "real" world. The "truth" determined by "stipulation"- suppose The book is on the table is true. Not the "real" world.The "truth" determined by "stipulation"- suppose The book is on the table is true. Not the "real" world. Mostly we are stipulate truth of a proposition. The truth value of P is true/false.Mostly we are stipulate truth of a proposition. The truth value of P is true/false. Fall 2013Frank / van Deemter / Wyner14

The Conjunction Operator The binary conjunction operator “  ” (AND) combines two propositions to form their logical conjunction. E.g. If p= “ I will have salad for lunch. ” and q= “ I will have steak for dinner. ”, then p  q= “ I will have salad for lunch and I will have steak for dinner. ” Fall 2013Frank / van Deemter / Wyner15

Note that a conjunction p 1  p 2  …  p n of n propositions will have 2 n rows in its truth table.Note that a conjunction p 1  p 2  …  p n of n propositions will have 2 n rows in its truth table. Also: ¬ and  operations together can express any Boolean truth table!Also: ¬ and  operations together can express any Boolean truth table! Conjunction Truth Table Operand columns Fall 2013Frank / van Deemter / Wyner16 more later

The Disjunction Operator The binary disjunction operator “  ” (OR) combines two propositions to form their logical disjunction. p= “ My car has a bad engine. ” q= “ My car has a bad carburator. ” p  q= “ Either my car has a bad engine, or my car has a bad carburetor. ” Fall 2013Frank / van Deemter / Wyner17

Note that p  q means that p is true, or q is true, or both are true!Note that p  q means that p is true, or q is true, or both are true! So, this operation is also called inclusive or, because it includes the possibility that both p and q are true.So, this operation is also called inclusive or, because it includes the possibility that both p and q are true. “ ¬ ” and “  ” together are also universal.“ ¬ ” and “  ” together are also universal. Disjunction Truth Table Note difference from AND Fall 2013Frank / van Deemter / Wyner18

Nested Propositional Expressions Use parentheses to group sub-expressions: “ I just saw my old friend, and either he ’ s grown or I ’ ve shrunk. ” = f  (g  s)Use parentheses to group sub-expressions: “ I just saw my old friend, and either he ’ s grown or I ’ ve shrunk. ” = f  (g  s) (f  g)  s would mean something different f  g  s would be ambiguous By convention, “ ¬ ” takes precedence over both “  ” and “  ”.By convention, “ ¬ ” takes precedence over both “  ” and “  ”. ¬s  f means (¬s)  f rather than ¬ (s  f) Fall 2013Frank / van Deemter / Wyner19

Logic as shorthand for NL Let p= “ It rained last night ”, q= “ The sprinklers came on last night, ” r= “ The lawn was wet this morning. ” ¬p = r  ¬p = ¬ r  p  q = It didn't rain last night. The lawn was wet this morning, and it didn ’ t rain last night. Either the lawn wasn ' t wet this morning, or it rained last night, or the sprinklers came on last night. Fall 2013Frank / van Deemter / Wyner20

Some important ideas: Distinguishing between different kinds of formulasDistinguishing between different kinds of formulas Seeing that some formulas that look different may express the same informationSeeing that some formulas that look different may express the same information First: different kinds of formulasFirst: different kinds of formulas Fall 2013Frank / van Deemter / Wyner21

Tautologies A tautology is a compound proposition that is true no matter what the truth values of its atomic propositions are! Ex. p   p [What is its truth table?] Fall 2013Frank / van Deemter / Wyner22

Tautologies When every row of the truth table gives.When every row of the truth table gives T. Example: p   p T T FT F T TFExample: p   p T T FT F T TF Fall 2013Frank / van Deemter / Wyner23

Contradictions A contradiction is a compound proposition that is false no matter what! Ex. p   p [Truth table?] Fall 2013Frank / van Deemter / Wyner24

Contradictions When every row of the truth table gives FWhen every row of the truth table gives F Example: p   p T F FT F F TFExample: p   p T F FT F F TF Fall 2013Frank / van Deemter / Wyner25

Contingencies All other props. are contingencies: Some rows give T, others give F Some rows give T, others give F Now: formulas that have the same meaning Fall 2013Frank / van Deemter / Wyner26

Propositional Equivalence Two syntactically (i.e., textually) different compound propositions may be semantically identical (i.e., have the same meaning). Here semantically identical means just that they have the same truth table for input truth values of the propositions. We call them logically equivalent. Notation: …  … Fall 2013Frank / van Deemter / Wyner27

Logical Equivalence Compound proposition p is logically equivalent to compound proposition q, written p  q, IFF p and q contain the same truth values in all rows of their truth tables We will also say: they express the same truth function (= the same function from values for atoms to values for the whole formula). Fall 2013Frank / van Deemter / Wyner28

Ex. Prove that p  q   (  p   q). Proving Equivalence via Truth Tables F T T T T T T T T T F F F F F F F F T T Fall 2013Frank / van Deemter / Wyner29 Shows that OR is equivalent to a combination of NOT and AND.

Before introducing more connectives … let us step back and ask a few questions about truth tables… let us step back and ask a few questions about truth tables Fall 2013Frank / van Deemter / Wyner30

1.What does each line of the table "mean"? 2.Consider a conjunction p 1  p 2  p 3 How many rows are there in its truth table? 3.Consider a conjunction p 1  p 2  …  p n of n propositions. How many rows are there in its truth table? 4.Explain why ¬ and  together are sufficient to express any Boolean truth table Questions for you to think about Fall 2013Frank / van Deemter / Wyner31

1.Consider a conjunction p 1  p 2  p 3 How many rows are there in its truth table? 8 p 1  p 2  p 3 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 Questions for you to think about Fall 2013Frank / van Deemter / Wyner32 Two truth values (0,1) and three propositions: 2 3 = 8.

2. Consider p 1  p 2  …  p n How many rows are there in its truth table? 2*2*2* … *2 (n factors) Hence 2 n (This grows exponentially!) Questions for you to think about Fall 2013Frank / van Deemter / Wyner33

3. Explain why ¬ and  together are sufficient to express any other complex expression in propositional logic. Questions for you to think about Fall 2013Frank / van Deemter / Wyner34

3.Explain why ¬ and  together are sufficient to express any other complex expression in propositional logic. Obviously, if we add new connectives (like  ) we can write new formulas.Obviously, if we add new connectives (like  ) we can write new formulas. CLAIM: these formulas would always be equivalent with ones that only use ¬ and  (This is what we need to prove).CLAIM: these formulas would always be equivalent with ones that only use ¬ and  (This is what we need to prove). Questions for you to think about Fall 2013Frank / van Deemter / Wyner35

Saying this in a different way: if we add new connectives, we can write new formulas, but these formulas will always only express truth functions that can already be expressed by formulas that only use ¬ and .Saying this in a different way: if we add new connectives, we can write new formulas, but these formulas will always only express truth functions that can already be expressed by formulas that only use ¬ and . That is, they will be equivalent.That is, they will be equivalent. Example of writing a disjunction in another form (equivalence shown before): p  q  ¬(¬p  ¬q)Example of writing a disjunction in another form (equivalence shown before): p  q  ¬(¬p  ¬q) Relating AND and OR Fall 2013Frank / van Deemter / Wyner36

Mystery Operator PQR Formula (containing P,Q,R) 1 1 1 0 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 Does there exist such a Formula? Fall 2013Frank / van Deemter / Wyner37 Suppose, given the truth values of P, Q, and R, we construct a Formula with the given resulting truth value. This is our 'mystery' operator. Can it be written equivalently with NOT and AND.

3. Explain why ¬ and  together are sufficient to express any Boolean truth table Suppose precisely two rows give T. For example, the rows whereSuppose precisely two rows give T. For example, the rows where –P=T, Q=T, R=F. This is P  Q  ¬R –P=T, Q=F, R=T. This is P  ¬Q  R T-values in Conjunction Fall 2013Frank / van Deemter / Wyner38

Suppose precisely two rows give T. For example, the rows where –P=T, Q=T, R=F. This is P  Q  ¬R –P=T, Q=F, R=T. This is P  ¬Q  R We’ve proven our claim if we can express the disjunction of these two rows: (P  Q  ¬R)  (P  ¬Q  R)We’ve proven our claim if we can express the disjunction of these two rows: (P  Q  ¬R)  (P  ¬Q  R) Table as a disjunction of T-rows Fall 2013Frank / van Deemter / Wyner39

We’ve arrived if we can express their disjunction: (P  Q  ¬R)  (P  ¬Q  R)We’ve arrived if we can express their disjunction: (P  Q  ¬R)  (P  ¬Q  R) But we’ve seen that disjunction can be expressed using  and ¬: A  B  ¬(¬A  ¬B)But we’ve seen that disjunction can be expressed using  and ¬: A  B  ¬(¬A  ¬B) So: (P  Q  ¬R)  (P  ¬Q  R)  ¬(¬ (P  Q  ¬R)   (P  ¬Q  R))So: (P  Q  ¬R)  (P  ¬Q  R)  ¬(¬ (P  Q  ¬R)   (P  ¬Q  R)) We’ve only used  and .We’ve only used  and . Disjoining rows of the table Fall 2013Frank / van Deemter / Wyner40

(P  Q  ¬R)  (P  ¬Q  R)  ¬(¬ (P  Q  ¬R)   (P  ¬Q  R)) 1 1 0 01 0 1 01 0 1 0 1 1 1 0 01 1 1 1 01 0 1 1 1 0 01 0 1 01 0 1 0 1 1 1 0 01 1 1 1 01 0 1 1 1 1 10 1 1 01 0 0 1 0 1 1 1 10 0 1 1 01 0 0 1 1 1 10 1 1 01 0 0 1 0 1 1 1 10 0 1 1 01 0 0 1 0 0 01 1 1 10 1 1 1 1 1 0 0 01 0 0 1 10 1 1 1 0 0 01 1 1 10 1 1 1 1 1 0 0 01 0 0 1 10 1 1 1 0 0 10 0 1 10 0 0 0 1 1 0 0 10 1 1 1 10 0 0 1 0 0 10 0 1 10 0 0 0 1 1 0 0 10 1 1 1 10 0 0 0 1 0 01 0 0 01 0 1 0 1 0 1 0 01 1 1 0 01 0 1 0 1 0 01 0 0 01 0 1 0 1 0 1 0 01 1 1 0 01 0 1 0 1 0 10 0 0 01 0 0 0 1 0 1 0 10 1 1 0 01 0 0 0 1 0 10 0 0 01 0 0 0 1 0 1 0 10 1 1 0 01 0 0 0 0 0 01 0 0 10 0 1 0 1 0 0 0 01 1 1 0 10 0 1 0 0 0 01 0 0 10 0 1 0 1 0 0 0 01 1 1 0 10 0 1 0 0 0 10 0 0 10 0 0 0 1 0 0 0 10 1 1 0 10 0 0 0 0 0 10 0 0 10 0 0 0 1 0 0 0 10 1 1 0 10 0 0 Check Fall 2013Frank / van Deemter / Wyner41 Notice that we only state where the new mystery connective is true as it is false elsewhere.

About this proof … We’ve made our task a bit easier, assuming that there were only 2 rows resulting in TWe’ve made our task a bit easier, assuming that there were only 2 rows resulting in T But the case with 1 or 3 or 4 or …. rows is analogous (and there are always only finitely many rows.)But the case with 1 or 3 or 4 or …. rows is analogous (and there are always only finitely many rows.) So, the proof can be made preciseSo, the proof can be made precise Fall 2013Frank / van Deemter / Wyner42

Having proved this … We can express every possible truth-functional operator in propositional logic in terms of AND and NOTWe can express every possible truth-functional operator in propositional logic in terms of AND and NOT This is sometimes called functional completeness. Also universality.This is sometimes called functional completeness. Also universality. Reduce other operators to other more basic operators.Reduce other operators to other more basic operators. Very useful in computing to reduce complexity of formulas (Normal forms)Very useful in computing to reduce complexity of formulas (Normal forms) Fall 2013Frank / van Deemter / Wyner43

Let’s introduce some additional connectives A variant of disjunctionA variant of disjunction The conditionalThe conditional The biconditionalThe biconditional Fall 2013Frank / van Deemter / Wyner44

The Exclusive Or Operator The binary exclusive-or operator “  ” (XOR) combines two propositions to form their logical “ exclusive or ”. p = “ I will earn an A in this course, ” q = “ I will drop this course, ” p  q = “ I will either earn an A in this course, or I will drop it (but not both!) ” Fall 2013Frank / van Deemter / Wyner45

Note that p  q means that p is true, or q is true, but not both!Note that p  q means that p is true, or q is true, but not both! This operation is called exclusive or, because it excludes the possibility that both p and q are true.This operation is called exclusive or, because it excludes the possibility that both p and q are true. “ ¬ ” and “  ” together are not universal.“ ¬ ” and “  ” together are not universal. Exclusive-Or Truth Table Note difference from OR. Fall 2013Frank / van Deemter / Wyner46

Note that English “ or ” can be ambiguous regarding the “ both ” case! Need context to disambiguate the meaning! For this class, assume “ or ” means inclusive. Natural Language is Ambiguous Fall 2013Frank / van Deemter / Wyner47

Test your understanding of the two types of disjunction 1.Suppose p  q is true. Does it follow that p  q is true? 2.Suppose p  q is true. Does it follow that p  q is true? Fall 2013Frank / van Deemter / Wyner48

Test your understanding of the two types of disjunction 1.Suppose p  q is true. Does it follow that p  q is true? No: consider p TRUE, q TRUE 2.Suppose p  q is true. Does it follow that p  q is true? Yes. Check each of the two assignments that make p  q true: a) p TRUE, q FALSE (makes p  q true) b) p FALSE, q TRUE (makes p  q true) Fall 2013Frank / van Deemter / Wyner49

The Implication Operator The implication p  q states that p implies q. I.e., If p is true, then q is true; but if p is not true, then q could be either true or false. E.g., let p = “ You study hard. ” q = “ You will get a good grade. ” p  q = “ If you study hard, then you will get a good grade. ” antecedentconsequent Fall 2013Frank / van Deemter / Wyner50

Implication Truth Table p  q is false only when (p is true but q is not true)p  q is false only when (p is true but q is not true) p  q does not say that p causes q!p  q does not say that p causes q! p  q does not require that p or q are true!p  q does not require that p or q are true! E.g. “ (1=0)  pigs can fly ” is TRUE!E.g. “ (1=0)  pigs can fly ” is TRUE! The only False case! Fall 2013Frank / van Deemter / Wyner51

Implication Truth Table Suppose you know that q is T. What do you know about p  q ?Suppose you know that q is T. What do you know about p  q ? Fall 2013Frank / van Deemter / Wyner52

Implication Truth Table Suppose you know that q is T. What do you know about p  q ?Suppose you know that q is T. What do you know about p  q ? The conditional must be TThe conditional must be T Fall 2013Frank / van Deemter / Wyner53

Implication Truth Table Suppose you know that p is F. What do you know about p  q ?Suppose you know that p is F. What do you know about p  q ? The conditional must be TThe conditional must be T Fall 2013Frank / van Deemter / Wyner54

Implication Truth Table Suppose you know that p is T.Suppose you know that p is T. What do you know about p  q ? T or F.What do you know about p  q ? T or F. What do you know aboutWhat do you know about q? T or F. q? T or F. Fall 2013Frank / van Deemter / Wyner55

Implications between real sentencs “ If this lecture ever ends, then the sun has risen this morning. ” True or False?“ If this lecture ever ends, then the sun has risen this morning. ” True or False? “ If Tuesday is a day of the week, then I am a penguin. ” True or False?“ If Tuesday is a day of the week, then I am a penguin. ” True or False? “ If 1+1=6, then Bush is president. ” True or False?“ If 1+1=6, then Bush is president. ” True or False? “ If the moon is made of green cheese, then 1+1=7. ” True or False?“ If the moon is made of green cheese, then 1+1=7. ” True or False? Fall 2013Frank / van Deemter / Wyner56

Why does this seem wrong? Recall “If [you study hard] then [you’ll get a good grade]”Recall “If [you study hard] then [you’ll get a good grade]” In normal English, this asserts a causal connection between the two propositions. The connective  does not capture this connection.In normal English, this asserts a causal connection between the two propositions. The connective  does not capture this connection. Fall 2013Frank / van Deemter / Wyner57

English Phrases Meaning p  q “ p implies q ”“ p implies q ” “ if p, then q ”“ if p, then q ” “ if p, q ”“ if p, q ” “ when p, q ”“ when p, q ” “ whenever p, q ”“ whenever p, q ” “ q if p ”“ q if p ” “ q when p ”“ q when p ” “ q whenever p ”“ q whenever p ” “ p only if q ” “ p is sufficient for q ” “ q is necessary for p ” “ q follows from p ” “ q is implied by p ” Fall 2013Frank / van Deemter / Wyner58 Tricky: "only if" is p  q. "p only if q" says that p cannot be true when q is not true. The statement "p only if q" is false where p is true, but q is false. Where p is true, then so is q.

But we’re not studying English.. Probably no two of these expressions have exactly the same meaning in EnglishProbably no two of these expressions have exactly the same meaning in English For example, ‘I’ll go to the party if Mary goes’ can be interpreted as implying ‘I’ll only go to the party if Mary goes’ turning the sentence into a biconditional: I go IFF Mary goesFor example, ‘I’ll go to the party if Mary goes’ can be interpreted as implying ‘I’ll only go to the party if Mary goes’ turning the sentence into a biconditional: I go IFF Mary goes Fall 2013Frank / van Deemter / Wyner59

Biconditional Truth Table p q means that p and q have the same truth value.p  q means that p and q have the same truth value. Note this truth table is the exact opposite of  ’ s!Note this truth table is the exact opposite of  ’ s! Thus, p q means ¬(p  q) Thus, p  q means ¬(p  q) p q does not imply that p and q are true, or that either of them causes the other.p  q does not imply that p and q are true, or that either of them causes the other. Fall 2013Frank / van Deemter / Wyner60

Consider... The truth of p q, where The truth of p  q, where 1.p= Scotland is in the UK q= 2+2 =4 2.p= Scotland is not in the UK q= 2+2 =5 3.p= Scotland is in the UK q= Wales is not in the UK Fall 2013Frank / van Deemter / Wyner61

Consider... The truth of p q, where The truth of p  q, where 1.p= Scotland is in the UK q= 2+2 =4 TRUE 2.p= Scotland is not in the UK q= 2+2 =5 TRUE 3.p= Scotland is in the UK q= Wales is not in the UK FALSE Fall 2013Frank / van Deemter / Wyner62

The difference between  and  A B says that A and B happen to have the same truth value. They could be atomic. A  B says that A and B happen to have the same truth value. They could be atomic. A  B says that no assignment of truth values to A and B can make A B false A  B says that no assignment of truth values to A and B can make A  B false So, A  B can only hold between well- chosen compound A and B. So, A  B can only hold between well- chosen compound A and B. A  B says “A B is a tautology” A  B says “A  B is a tautology” Fall 2013Frank / van Deemter / Wyner63

Consider... The truth of p  q, where 1.p= Scotland is in the UK q= 2+2 =4 2.p= Scotland is not in the UK q= Wales is not in the UK 3.p= Scotland is in UK  Wales is in UK q= Wales is in UK  Scotland is in UK Fall 2013Frank / van Deemter / Wyner64

Consider... The truth of p  q, where 1.p= Scotland is in the UK q= 2+2 =4 FALSE 2.p= Scotland is not in the UK q= Wales is not in the UK FALSE 3.p= Scotland is in UK  Wales is in UK q= Wales is in UK  Scotland is in UK TRUE Fall 2013Frank / van Deemter / Wyner65

The language of propositional logic defined more properly (i.e., as a formal language) Atoms: p1, p2, p3,..Atoms: p1, p2, p3,.. Formulas:Formulas: –All atoms are formulas –For all , if  is a formula then ¬  is a formula –For all  and , if  and  are formulas then the following are formulas: (    ), (    ), (    ) (etc.) Fall 2013Frank / van Deemter / Wyner66

The language of propositional logic defined more properly (i.e., as a formal language) Which of these are formulas, according to this strict definition?Which of these are formulas, according to this strict definition? p1  ¬ p2 p1  ¬ p2 (p1  ¬ p2)(p1  ¬ p2) ¬ ¬ ¬(p9  p8) ¬ ¬ ¬(p9  p8) (p1  p2  p3)(p1  p2  p3) (p1  (p2  p3))(p1  (p2  p3)) Fall 2013Frank / van Deemter / Wyner67

The language of propositional logic defined more properly (i.e., as a formal language) Which of these are formulas, according to this strict definition?Which of these are formulas, according to this strict definition? p1  ¬ p2 No p1  ¬ p2 No (p1  ¬ p2) Yes(p1  ¬ p2) Yes ¬ ¬ ¬(p9  p8) Yes ¬ ¬ ¬(p9  p8) Yes (p1  p2  p3) No(p1  p2  p3) No (p1  (p2  p3)) Yes(p1  (p2  p3)) Yes Fall 2013Frank / van Deemter / Wyner68

Simplifying conventions Convention 1: outermost brackets may be omitted,: p1  ¬ p2, ¬ ¬ ¬(p9  p8), p1  (p2  p3)Convention 1: outermost brackets may be omitted,: p1  ¬ p2, ¬ ¬ ¬(p9  p8), p1  (p2  p3) Convention 2: associativity allows us to omit even more brackets, e.g.: p1  p2  p3, p1  p2  p3Convention 2: associativity allows us to omit even more brackets, e.g.: p1  p2  p3, p1  p2  p3 Fall 2013Frank / van Deemter / Wyner69

The language of propositional logic defined more properly (i.e., as a formal language) Which of these are formulas, when using these two conventions?Which of these are formulas, when using these two conventions? p1  ¬ p2 Yes p1  ¬ p2 Yes (p1  ¬ p2) Yes(p1  ¬ p2) Yes ¬ ¬ ¬(p9  p8) Yes ¬ ¬ ¬(p9  p8) Yes (p1  p2  p3) No(p1  p2  p3) No (p1  (p2  p3)) Yes(p1  (p2  p3)) Yes Fall 2013Frank / van Deemter / Wyner70

Contrapositive Some terminology, for an implication p  q: Its converse is: q  p.Its converse is: q  p. Its contrapositive:¬q  ¬ p.Its contrapositive:¬q  ¬ p. Which of these two has/have the same meaning (express same truth function) as p  q? Prove it.Which of these two has/have the same meaning (express same truth function) as p  q? Prove it. Fall 2013Frank / van Deemter / Wyner71

Contrapositive Proving the equivalence of p  q and its contrapositive, using truth tables: Fall 2013Frank / van Deemter / Wyner72

Boolean Operations Summary We have seen 1 unary operator (out of the 4 possible ones) and 5 binary operators:We have seen 1 unary operator (out of the 4 possible ones) and 5 binary operators: Fall 2013Frank / van Deemter / Wyner73

For you to think about: 1.Can you think of yet another 2-place connective? How many possible connectives do there exist? Fall 2013Frank / van Deemter / Wyner74

For you to think about: 1.How many possible connectives do there exist? p connective q T ? T T ? F F ? T F ? F Each question mark can be T or F, hence 2*2*2*2=16 connectives p connective q T ? T T ? F F ? T F ? F Each question mark can be T or F, hence 2*2*2*2=16 connectives Fall 2013Frank / van Deemter / Wyner75

Example of another connective p connective q compare: p and q p connective q compare: p and q T F T T T T F F F T T F F T F F T F T T T T F F F T T F F T F F Names: NAND, Sheffer stroke Fall 2013Frank / van Deemter / Wyner76

Some Alternative Notations Fall 2013Frank / van Deemter / Wyner77

To this point You have learned about: Propositional logic operators ’Propositional logic operators ’ –Symbolic notations. –English equivalents –Truth tables. –Logical equivalence Next: –More about logical equivalences. –How to prove them. Fall 2013Frank / van Deemter / Wyner78

Tautologies revisited We’ve introduced the notion of a tautology using the example p   pWe’ve introduced the notion of a tautology using the example p   p Now, you know more operators, so can formulate many more tautologies, e.g., the following are tautologies: (p  q)   (  p   q) (p  q)  (¬q  ¬ p), and so onNow, you know more operators, so can formulate many more tautologies, e.g., the following are tautologies: (p  q)   (  p   q) (p  q)  (¬q  ¬ p), and so on Fall 2013Frank / van Deemter / Wyner79

Equivalence Laws Similar to arithmetic identities in algebraSimilar to arithmetic identities in algebra Patterns that can be used to match (part of) another propositionPatterns that can be used to match (part of) another proposition Abbreviation: T stands for an arbitrary tautology; F an arbitrary contradictionAbbreviation: T stands for an arbitrary tautology; F an arbitrary contradiction Fall 2013Frank / van Deemter / Wyner80

Equivalence Laws - Examples Identity: p  T  p p  F  pIdentity: p  T  p p  F  p Domination: p  T  T p  F  FDomination: p  T  T p  F  F Idempotence: p  p  p p  p  pIdempotence: p  p  p p  p  p Double negation:  p  pDouble negation:  p  p Commutativity: p  q  q  p p  q  q  pCommutativity: p  q  q  p p  q  q  p Associativity: (p  q)  r  p  (q  r) (p  q)  r  p  (q  r)Associativity: (p  q)  r  p  (q  r) (p  q)  r  p  (q  r) Fall 2013Frank / van Deemter / Wyner81

More Equivalence Laws Distributive: p  (q  r)  (p  q)  (p  r) p  (q  r)  (p  q)  (p  r)Distributive: p  (q  r)  (p  q)  (p  r) p  (q  r)  (p  q)  (p  r) De Morgan ’ s:  (p  q)   p   q  (p  q)   p   qDe Morgan ’ s:  (p  q)   p   q  (p  q)   p   q Trivial tautology/contradiction: p   p  T p   p  FTrivial tautology/contradiction: p   p  T p   p  F Augustus De Morgan (1806-1871) Fall 2013Frank / van Deemter / Wyner82

Defining Operators via Equivalences Some equivalences can be thought of as definitions of one operator in terms of others: Exclusive or: p  q  (p  q)  (p  q) p  q  (p  q)  (q  p)Exclusive or: p  q  (p  q)  (p  q) p  q  (p  q)  (q  p) Implies: p  q   p  qImplies: p  q   p  q Biconditional: p  q  (p  q)  (q  p) p  q   (p  q)Biconditional: p  q  (p  q)  (q  p) p  q   (p  q) Fall 2013Frank / van Deemter / Wyner83

How you may use equivalence laws: Example (1) Use equivalences to prove that  (  r   s)  s  r.Use equivalences to prove that  (  r   s)  s  r. Fall 2013Frank / van Deemter / Wyner84

How you may use equivalence laws: Example (1)  (  r   s)  (De Morgan)  (  r   s)  (De Morgan)  r   s  (Commutativity)  s   r  (2x Double Negation) s  r.s  r.s  r.s  r. Fall 2013Frank / van Deemter / Wyner85

How you may use equivalence laws: Example (2) Use equivalences to prove that (p   q)  (p  r)   p  q   r.Use equivalences to prove that (p   q)  (p  r)   p  q   r. (This would be much easier using truth tables!)(This would be much easier using truth tables!) (p   q)  (p  r) [Expand “ definition ” of  ]   (p   q)  (p  r) [Expand “ definition ” of  ]   (p   q)  (p  r) [Expand “ definition ” of  ]   (p   q)  ((p  r)   (p  r)) [DeMorgan]  (  p  q)  ((p  r)   (p  r)) cont. cont. Fall 2013Frank / van Deemter / Wyner86 Underline just focuses interest.

Long example Continued... (  p  q)  ((p  r)   (p  r))  [  commutes]  (q   p)  ((p  r)   (p  r)) [  is associative]  q  (  p  ((p  r)   (p  r))) [distrib.  over  ]  q  (((  p  (p  r))  (  p   (p  r))) [assoc.]  q  (((  p  p)  r)  (  p   (p  r))) [taut.]  q  ((T  r)  (  p   (p  r))) [domination]  q  (T  (  p   (p  r))) [identity]  q  (  p   (p  r))  cont. Fall 2013Frank / van Deemter / Wyner87

End of Long Example q  (  p   (p  r)) [DeMorgan]  q  (  p  (  p   r)) [Assoc.]  q  ((  p   p)   r) [Idempotent]  q  (  p   r) [Associativity]  (q   p)   r [Commutativity]   p  q   r Q.E.D. (quod erat demonstrandum) Fall 2013Frank / van Deemter / Wyner88

Summary In practice, Propositional Logic equivalences are seldom strung together into long proofs: using truth tables is usually much easier.In practice, Propositional Logic equivalences are seldom strung together into long proofs: using truth tables is usually much easier. We shall now turn to a more complex logic, where nothing like truth tables is available.We shall now turn to a more complex logic, where nothing like truth tables is available. Fall 2013Frank / van Deemter / Wyner89

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