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Fall 2002 CMSC 203 - Discrete Structures 1

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LogicLogic!!


LogicLogic• Crucial for mathematical reasoningCrucial for mathematical reasoning• Used for designing electronic circuitryUsed for designing electronic circuitry

• Logic is a system based on Logic is a system based on propositionspropositions..• A proposition is a statement that is either A proposition is a statement that is either

truetrue or or falsefalse (not both). (not both).• We say that the We say that the truth valuetruth value of a of a

proposition is either true (T) or false (F).proposition is either true (T) or false (F).

• Corresponds to Corresponds to 11 and and 00 in digital circuits in digital circuits


The Statement/Proposition The Statement/Proposition GameGame

““Elephants are bigger than mice.”Elephants are bigger than mice.”

Is this a statement?Is this a statement? yesyes

Is this a proposition?Is this a proposition? yesyes

What is the truth What is the truth value value of the proposition?of the proposition?

truetrue



““520 < 111”520 < 111”

Is this a statement?Is this a statement? yesyes



falsfalsee



““y > 5”y > 5”Is this a statement?Is this a statement? yesyesIs this a proposition?Is this a proposition? nonoIts truth value depends on the value of Its truth value depends on the value of y, but this value is not specified.y, but this value is not specified.We call this type of statement a We call this type of statement a propositional functionpropositional function or or open open sentencesentence..



““Today is January 1 and 99 < 5.”Today is January 1 and 99 < 5.”Is this a statement?Is this a statement? yesyes



falsfalsee



““Please do not fall asleep.”Please do not fall asleep.”Is this a statement?Is this a statement? nono

Is this a proposition?Is this a proposition? nonoOnly statements can be propositions.Only statements can be propositions.

It’s a request.It’s a request.



““If elephants were red,If elephants were red,they could hide in cherry trees.”they could hide in cherry trees.”

Is this a statement?Is this a statement? yesyesIs this a proposition?Is this a proposition? yesyes


probably probably falsefalse



““x < y if and only if y > x.”x < y if and only if y > x.”Is this a statement?Is this a statement? yesyesIs this a proposition?Is this a proposition? yesyes


truetrue

… … because its truth value because its truth value does not depend on does not depend on specific values of x and specific values of x and y.y.


Combining PropositionsCombining Propositions

As we have seen in the previous examples, As we have seen in the previous examples, one or more propositions can be combined one or more propositions can be combined to form a single to form a single compound propositioncompound proposition..

We formalize this by denoting propositions We formalize this by denoting propositions with letters such as with letters such as p, q, r, s,p, q, r, s, and and introducing several introducing several logical operatorslogical operators. .


Logical Operators Logical Operators (Connectives)(Connectives)

We will examine the following logical We will examine the following logical operators:operators:• Negation Negation (NOT)(NOT)• Conjunction Conjunction (AND)(AND)• Disjunction Disjunction (OR)(OR)• Exclusive or Exclusive or (XOR)(XOR)• Implication Implication (if – then)(if – then)• Biconditional Biconditional (if and only if)(if and only if)Truth tables can be used to show how these Truth tables can be used to show how these operators can combine propositions to operators can combine propositions to compound propositions.compound propositions.


Negation (NOT)Negation (NOT)Unary Operator, Symbol: Unary Operator, Symbol:

PP PPtrue (T)true (T) false (F)false (F)false (F)false (F) true (T)true (T)


Conjunction (AND)Conjunction (AND)Binary Operator, Symbol: Binary Operator, Symbol:

PP QQ PPQQTT TT TTTT FF FFFF TT FFFF FF FF


Disjunction (OR)Disjunction (OR)Binary Operator, Symbol: Binary Operator, Symbol:

PP QQ PPQQTT TT TTTT FF TTFF TT TTFF FF FF


Exclusive Or (XOR)Exclusive Or (XOR)Binary Operator, Symbol: Binary Operator, Symbol:

PP QQ PPQQTT TT FFTT FF TTFF TT TTFF FF FF


Implication (if - then)Implication (if - then)Binary Operator, Symbol: Binary Operator, Symbol:

PP QQ PPQQTT TT TTTT FF FFFF TT TTFF FF TT


Biconditional (if and only Biconditional (if and only if)if)

Binary Operator, Symbol: Binary Operator, Symbol: PP QQ PPQQTT TT TTTT FF FFFF TT FFFF FF TT


Statements and OperatorsStatements and OperatorsStatements and operators can be combined in Statements and operators can be combined in

any way to form new statements.any way to form new statements.

PP QQ PP QQ ((P)P)((Q)Q)TT TT FF FF FFTT FF FF TT TTFF TT TT FF TTFF FF TT TT TT


Statements and Statements and OperationsOperations

Statements and operators can be combined in Statements and operators can be combined in any way to form new statements.any way to form new statements.

PP QQ PPQQ (P(PQ)Q) ((P)P)((Q)Q)TT TT TT FF FFTT FF FF TT TTFF TT FF TT TTFF FF FF TT TT


Equivalent StatementsEquivalent StatementsPP QQ (P(PQ)Q) ((P)P)((Q)Q) (P(PQ)Q)((P)P)((Q)Q)

TT TT FF FF TTTT FF TT TT TTFF TT TT TT TTFF FF TT TT TT

The statements The statements (P(PQ) and (Q) and (P) P) ( (Q) are Q) are logically logically equivalentequivalent, since , since (P(PQ) Q) ((P) P) ( (Q) is always true.Q) is always true.


Tautologies and Tautologies and ContradictionsContradictions

A tautology is a statement that is always true.A tautology is a statement that is always true.Examples: Examples: • RR((R)R) (P(PQ)Q)((P)P)((Q)Q)

If SIf ST is a tautology, we write ST is a tautology, we write ST.T.If SIf ST is a tautology, we write ST is a tautology, we write ST.T.



A contradiction is a statement that is alwaysA contradiction is a statement that is alwaysfalse.false.Examples: Examples: • RR((R)R) (((P(PQ)Q)((P)P)((Q))Q))The negation of any tautology is a contra-The negation of any tautology is a contra-diction, and the negation of any contradiction is diction, and the negation of any contradiction is a tautology.a tautology.


ExercisesExercisesWe already know the following tautology: We already know the following tautology: (P(PQ) Q) ((P)P)((Q)Q)Nice home exercise: Nice home exercise: Show that Show that (P(PQ) Q) ((P)P)((Q).Q).These two tautologies are known as De These two tautologies are known as De Morgan’s laws.Morgan’s laws.Table 5 in Section 1.2Table 5 in Section 1.2 shows many useful laws. shows many useful laws.Exercises 1 and 7 in Section 1.2Exercises 1 and 7 in Section 1.2 may help you may help you get used to propositions and operators.get used to propositions and operators.


Let’s Talk About LogicLet’s Talk About Logic• Logic is a system based on Logic is a system based on propositionspropositions..

• A proposition is a statement that is either A proposition is a statement that is either truetrue or or falsefalse (not both). (not both).

• We say that the We say that the truth valuetruth value of a of a proposition is either true (T) or false (F).proposition is either true (T) or false (F).

• Corresponds to Corresponds to 11 and and 00 in digital circuits in digital circuits


Logical Operators Logical Operators (Connectives)(Connectives)

• Negation Negation (NOT)(NOT)• Conjunction Conjunction (AND)(AND)• Disjunction Disjunction (OR)(OR)• Exclusive or Exclusive or (XOR)(XOR)• Implication Implication (if – then)(if – then)• Biconditional Biconditional (if and only if)(if and only if)Truth tables can be used to show how these Truth tables can be used to show how these operators can combine propositions to operators can combine propositions to compound propositions.compound propositions.



A tautology is a statement that is always true.A tautology is a statement that is always true.Examples: Examples: • RR((R)R) (P(PQ)Q)((P)P)((Q)Q)

If SIf ST is a tautology, we write ST is a tautology, we write ST.T.If SIf ST is a tautology, we write ST is a tautology, we write ST.T.



A contradiction is a statement that is alwaysA contradiction is a statement that is alwaysfalse.false.Examples: Examples: • RR((R)R)• (((P(PQ)Q)((P)P)((Q))Q))

The negation of any tautology is a The negation of any tautology is a contradiction, and the negation of any contradiction, and the negation of any contradiction is a tautology.contradiction is a tautology.


Propositional FunctionsPropositional FunctionsPropositional function (open sentence):Propositional function (open sentence):statement involving one or more variables,statement involving one or more variables,e.g.: x-3 > 5.e.g.: x-3 > 5.Let us call this propositional function P(x), Let us call this propositional function P(x), where P is the where P is the predicatepredicate and x is the and x is the variablevariable..

What is the truth value of P(2) ?What is the truth value of P(2) ? falsefalseWhat is the truth value of P(8) ?What is the truth value of P(8) ?What is the truth value of P(9) ?What is the truth value of P(9) ?

falsefalsetruetrue


Propositional FunctionsPropositional FunctionsLet us consider the propositional function Let us consider the propositional function Q(x, y, z) defined as: Q(x, y, z) defined as: x + y = z.x + y = z.Here, Q is the Here, Q is the predicatepredicate and x, y, and z are and x, y, and z are the the variablesvariables..

What is the truth value of Q(2, 3, 5) What is the truth value of Q(2, 3, 5) ??

truetrueWhat is the truth value of Q(0, 1, What is the truth value of Q(0, 1, 2) ?2) ?What is the truth value of Q(9, -9, 0) ?What is the truth value of Q(9, -9, 0) ?

falsefalsetruetrue


Universal QuantificationUniversal QuantificationLet P(x) be a propositional function.Let P(x) be a propositional function.

Universally quantified sentenceUniversally quantified sentence::For all x in the universe of discourse P(x) is true.For all x in the universe of discourse P(x) is true.

Using the universal quantifier Using the universal quantifier ::x P(x) x P(x) “for all x P(x)” or “for every x P(x)”“for all x P(x)” or “for every x P(x)”

(Note: (Note: x P(x) is either true or false, so it is a x P(x) is either true or false, so it is a proposition, not a propositional function.)proposition, not a propositional function.)


Universal QuantificationUniversal QuantificationExample: Example: S(x): x is a UMBC student.S(x): x is a UMBC student.G(x): x is a genius.G(x): x is a genius.

What does What does x (S(x) x (S(x) G(x)) G(x)) mean ? mean ?

““If x is a UMBC student, then x is a genius.”If x is a UMBC student, then x is a genius.”oror““All UMBC students are geniuses.”All UMBC students are geniuses.”


Existential QuantificationExistential QuantificationExistentially quantified sentenceExistentially quantified sentence::There exists an x in the universe of discourse There exists an x in the universe of discourse for which P(x) is true.for which P(x) is true.

Using the existential quantifier Using the existential quantifier ::x P(x) x P(x) “There is an x such that P(x).”“There is an x such that P(x).”

“ “There is at least one x such that There is at least one x such that P(x).”P(x).”

(Note: (Note: x P(x) is either true or false, so it is a x P(x) is either true or false, so it is a proposition, but no propositional function.)proposition, but no propositional function.)


Existential QuantificationExistential QuantificationExample: Example: P(x): x is a UMBC professor.P(x): x is a UMBC professor.G(x): x is a genius.G(x): x is a genius.

What does What does x (P(x) x (P(x) G(x)) G(x)) mean ? mean ?

““There is an x such that x is a UMBC There is an x such that x is a UMBC professor and x is a genius.”professor and x is a genius.”oror““At least one UMBC professor is a genius.”At least one UMBC professor is a genius.”


QuantificationQuantificationAnother example:Another example:Let the universe of discourse be the real numbers.Let the universe of discourse be the real numbers.

What does What does xxy (x + y = 320)y (x + y = 320) mean ? mean ?

““For every x there exists a y so that x + y = 320.”For every x there exists a y so that x + y = 320.”

Is it true?Is it true?

Is it true for the natural Is it true for the natural numbers?numbers?

yesyes

nono


Disproof by CounterexampleDisproof by CounterexampleA counterexample to A counterexample to x P(x) is an object c so x P(x) is an object c so that P(c) is false. that P(c) is false.

Statements such as Statements such as x (P(x) x (P(x) Q(x)) can be Q(x)) can be disproved by simply providing a disproved by simply providing a counterexample.counterexample.

Statement: “All birds can fly.”Statement: “All birds can fly.”Disproved by counterexample: Penguin.Disproved by counterexample: Penguin.


NegationNegation

((x P(x)) is logically equivalent to x P(x)) is logically equivalent to x (x (P(x)).P(x)).

((x P(x)) is logically equivalent to x P(x)) is logically equivalent to x (x (P(x)).P(x)).

See Table 3 in Section 1.3.See Table 3 in Section 1.3.

I recommend exercises 5 and 9 in Section 1.3.I recommend exercises 5 and 9 in Section 1.3.

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