Symbolic LogicSymbolic Logic The Following slide were written using The Following slide were written using

materials from the Book:materials from the Book:

Discrete mathematics With Discrete mathematics With ApplicationsApplicationsThird EditionThird Edition

By Susanna S. EppBy Susanna S. Epp

Symbolic LogicSymbolic Logic The main purpose of logic is to buildThe main purpose of logic is to build the the

thinking methods. thinking methods. PProvide rules,techniques, rovide rules,techniques, for makingfor making

decision in decision in an an argument,argument, validating avalidating a deduction.deduction.

InIn classical logic, only phrases , classical logic, only phrases , assertionsassertions with one truthwith one truth value are allowed: TRUE or value are allowed: TRUE or FALSE, without ambiguityFALSE, without ambiguity

Such Such boolean assertions are called boolean assertions are called propositionspropositions. .

Symbolic LogicSymbolic Logic A A propositiopropositionn is a statement that is either true is a statement that is either true

oror false.false. The purpose of propositionalThe purpose of propositional logic is to logic is to

provide complex construction of rules, from provide complex construction of rules, from anonymousanonymous propositions calledpropositions called propositional propositional variables variables

If If pp is a proposition ,the is a proposition ,the negationnegation of p, of p, denoted by ¬denoted by ¬pp, is a proposition which means " , is a proposition which means " itit is false that". Then if is false that". Then if pp is true , ¬ is true , ¬pp is false, is false, andand if p is false , ¬p is true.if p is false , ¬p is true.

Symbolic LogicSymbolic Logic A propositionA proposition consisting of only a single consisting of only a single

propositional variable or singlepropositional variable or single constant constant (true or false) is called an atomic (true or false) is called an atomic proposition.proposition.

All nonatomic propositions are called All nonatomic propositions are called compoundcompound propositions. All compound propositions. All compound propositions contain at least onepropositions contain at least one logical logical connective. connective.

Symbolic LogicSymbolic Logic If If pp and and qq are propositions, the are propositions, the conjunctionconjunction of of pp

and and q q is the proposition " is the proposition " p and qp and q”” denoted by denoted by ppqq. .

The proposition The proposition ppqq is true if p and q are is true if p and q are both both true, and false otherwise; this is describe by the true, and false otherwise; this is describe by the followingfollowing truth table:truth table:

pp qq ppqq00 00 0000 11 0011 00 0011 11 11

Symbolic LogicSymbolic Logic TThe he disdisjunctionjunction of of pp and and q q is the proposition " is the proposition " p p

oror q q”” denoted by denoted by ppqq. . The proposition The proposition ppqq is true if at least one of is true if at least one of the the

two propositions two propositions pp and and pp is true, and false when is true, and false when pp and and qq are are both false; this is described by theboth false; this is described by the following truth table:following truth table:

pp qq ppqq00 00 0000 11 1111 00 1111 11 11

Symbolic LogicSymbolic Logic By combining,¬By combining,¬,,,, we can build we can build compound compound

propositions and construct their truth tables.propositions and construct their truth tables. Truth table for: (pTruth table for: (p q )q ) ¬¬rr

pp qq rr p p qq ¬¬rr (p(p q )q ) ¬¬rr11 11 11 11 00 1111 11 00 11 11 1111 00 11 00 00 0011 00 00 00 11 1100 11 11 00 00 0000 11 00 00 11 1100 00 11 00 00 0000 00 00 00 11 11

(p(p q )q ) ¬¬rr

Logical equivalenceLogical equivalence Two statements are Two statements are logically equivalentlogically equivalent if they if they

have equivalent truth tables.have equivalent truth tables. The symbol for The symbol for Logical equivalence is Logical equivalence is Example: p Example: p q q p

PP QQ p p qq q q pp11 11 11 11

11 00 00 00

00 11 00 00

00 00 00 00

Double negative propertyDouble negative property The negation of the negation of a statement The negation of the negation of a statement

is logically equivalent to the statementis logically equivalent to the statement ¬¬((¬¬p) p) p p

pp ¬¬pp ¬¬((¬¬p)p)

11 00 11

00 11 00

Showing nonequivalenceShowing nonequivalence Show that the statement forms Show that the statement forms ¬¬(p(pq) and q) and

¬¬pp¬¬q are not logically equivalent. q are not logically equivalent.

pp qq ¬¬pp ¬¬qq ppqq ¬¬(p(pq)q) ¬¬pp¬¬qq

11 11 00 00 11 00 00

11 00 00 11 00 11 00

00 11 11 00 00 11 00

00 00 11 11 00 11 11

De Morgan’s laws De Morgan’s laws The negation of a conjunction of two The negation of a conjunction of two

statements is logically equivalent to the statements is logically equivalent to the disjunction of their negations.disjunction of their negations.

¬¬(p(pq) q) ¬¬p p ¬¬qq The negation of the disjunction of two The negation of the disjunction of two

statements is logically equivalent to the statements is logically equivalent to the conjunction of their negation.conjunction of their negation.

¬¬(p (p q) q) ¬¬p p ¬¬qq

pp qq ¬¬pp ¬¬qq ppqq ¬¬(p(pq)q) ¬¬p p ¬¬qq

11 11 00 00 11 00 00

11 00 00 11 00 11 11

00 11 11 00 00 11 11

00 00 11 11 00 11 11

De Morgan’s laws De Morgan’s laws

pp qq ¬¬pp ¬¬qq ppqq ¬¬(p(pq)q) ¬¬pp¬¬qq

11 11 00 00 11 00 00

11 00 00 11 11 00 00

00 11 11 00 11 00 00

00 00 11 11 00 11 11

De Morgan’s laws De Morgan’s laws

Tautologies and ContradictionTautologies and Contradiction A A tautologytautology is a statement form that is is a statement form that is

always truealways true regardless of the truth values of regardless of the truth values of individual statements substituted for its individual statements substituted for its statement variables.statement variables.

A A contradictioncontradiction is a statement form that is is a statement form that is always falsealways false regardless of the truth values regardless of the truth values of individual statements substituted for its of individual statements substituted for its statement variables.statement variables.

Tautologies and ContradictionTautologies and Contradiction

pp ¬¬pp pp¬¬pp pp¬¬pp

11 00 11 00

00 11 11 00

Tautology Contradiction

Logical EquivalencesLogical Equivalences Given any statement variables Given any statement variables pp, , qq, and , and rr, a tautology , a tautology

tt and a contradiction and a contradiction cc the following logical the following logical equivalences hold.equivalences hold.

1.1. Commutative lawCommutative lawppq q q qp p p p q q q q pp2.2. AssociativeAssociative law law(p(pq)q)r r p p (q(qr ) r ) (p (pq)q)r r p p(q(qr ) r ) 3.3. Distributive lawDistributive lawpp(q(qr)r)(p(pq)q)(p(pr ) r ) pp(q (q r)r)(p(pq) q) (p(pr ) r ) 4.4. Identity Identity p p tt p p p p cc p p

Logical EquivalencesLogical Equivalences5.5. Negation lawsNegation lawspp¬¬pp t t p p ¬¬pp c c6.6. Double negative lawDouble negative law¬ ¬ ((¬ ¬ p)p)pp7.7. IdempotentIdempotentpppp p p p ppp p p8.8. Universal bounds lawsUniversal bounds lawspptt tt p pcc cc9.9. De Morgan’s lawsDe Morgan’s laws

¬¬(p(pq) q) ¬¬p p ¬¬qq ¬¬(p (p q) q) ¬¬p p ¬¬qq

Logical EquivalencesLogical Equivalences10.10. Absorption lawsAbsorption lawspp(p(pq) q) p p p p (p(pq) q) p p11.11. Negations of t and cNegations of t and c¬¬tt cc ¬¬cc tt

Simplifying statementsSimplifying statements Verify the equivalenceVerify the equivalence ¬¬((¬¬ppq)q)(p (p q) q) p p By De Morgan’sBy De Morgan’s ((¬¬((¬¬p)p)¬¬q)q)(p (p q)q) By double negative lawBy double negative law (p(p¬¬q)q)(p (p q)q) By distributive lawBy distributive law pp((¬¬qqq)q) By negation lawBy negation law ppcc By identity lawBy identity law pp

Simplifying statementsSimplifying statements Verify the equivalenceVerify the equivalence ¬¬(p(p¬¬q)q)((¬¬pp¬ ¬ q) q) ¬¬pp

Simplify the following expressions. State the rule Simplify the following expressions. State the rule you are using at each stage.you are using at each stage.

            ((((ppq)q) ( (ppq))q))(p(pq)q) (( ( (ppq)q) ( (ppq))q))(p(pq) De Morgan’s lawq) De Morgan’s law (( ( ()p)pq)q)(( ( ()p)p(()q)))q))(p(pq)q) De De

Morgan’s lawMorgan’s law (p(pq)q)(p(pq)q)(p(pq)q) Double negative lawDouble negative law pp ( (qqq)q)(p(pq)q) Distributive lawsDistributive laws (p(p tt ))(p(pq)q) Negation lawsNegation laws pp(p(pq)q) Universal bounds lawsUniversal bounds laws ppqq