Symbolic Logic The Following slide were written using materials from the Book: The Following slide were written using materials from the Book: Discrete mathematics With Applications Third Edition By Susanna S. Epp

Symbolic Logic The main purpose of logic is to build the thinking methods. The main purpose of logic is to build the thinking methods. Provide rules,techniques, for making decision in an argument, validating a deduction. Provide rules,techniques, for making decision in an argument, validating a deduction. In classical logic, only phrases, assertions with one truth value are allowed: TRUE or FALSE, without ambiguity In classical logic, only phrases, assertions with one truth value are allowed: TRUE or FALSE, without ambiguity Such boolean assertions are called propositions. Such boolean assertions are called propositions.

Symbolic Logic A proposition is a statement that is either true or false. A proposition is a statement that is either true or false. The purpose of propositional logic is to provide complex construction of rules, from anonymous propositions called propositional variables The purpose of propositional logic is to provide complex construction of rules, from anonymous propositions called propositional variables If p is a proposition,the negation of p, denoted by ¬p, is a proposition which means " it is false that". Then if p is true, ¬p is false, and if p is false, ¬p is true. If p is a proposition,the negation of p, denoted by ¬p, is a proposition which means " it is false that". Then if p is true, ¬p is false, and if p is false, ¬p is true.

Symbolic Logic A proposition consisting of only a single propositional variable or single constant (true or false) is called an atomic proposition. A proposition consisting of only a single propositional variable or single constant (true or false) is called an atomic proposition. All nonatomic propositions are called compound propositions. All compound propositions contain at least one logical connective. All nonatomic propositions are called compound propositions. All compound propositions contain at least one logical connective.

Symbolic Logic If p and q are propositions, the conjunction of p and q is the proposition " p and q” denoted by p  q. If p and q are propositions, the conjunction of p and q is the proposition " p and q” denoted by p  q. The proposition p  q is true if p and q are both true, and false otherwise; this is describe by the following truth table: The proposition p  q is true if p and q are both true, and false otherwise; this is describe by the following truth table: pq pqpqpqpq

Symbolic Logic The disjunction of p and q is the proposition " p or q” denoted by p  q. The disjunction of p and q is the proposition " p or q” denoted by p  q. The proposition p  q is true if at least one of the two propositions p and p is true, and false when p and q are both false; this is described by the following truth table: The proposition p  q is true if at least one of the two propositions p and p is true, and false when p and q are both false; this is described by the following truth table: pq pqpqpqpq

Symbolic Logic By combining,¬, ,  we can build compound propositions and construct their truth tables. By combining,¬, ,  we can build compound propositions and construct their truth tables. Truth table for: (p  q )  ¬r Truth table for: (p  q )  ¬r

pqr p  q ¬r¬r¬r¬r (p  q )  ¬r

Logical equivalence Two statements are logically equivalent if they have equivalent truth tables. Two statements are logically equivalent if they have equivalent truth tables. The symbol for Logical equivalence is  The symbol for Logical equivalence is  Example: p    Example: p  q  q  p PQ p  q q  p

Double negative property The negation of the negation of a statement is logically equivalent to the statement The negation of the negation of a statement is logically equivalent to the statement ¬(¬p)  p ¬(¬p)  p p ¬p¬p¬p¬p ¬(¬p)

Showing nonequivalence Show that the statement forms ¬(p  q) and ¬p  ¬q are not logically equivalent. Show that the statement forms ¬(p  q) and ¬p  ¬q are not logically equivalent. pq ¬p¬p¬p¬p ¬q¬q¬q¬q pqpqpqpq ¬(p  q) ¬p¬q¬p¬q¬p¬q¬p¬q

De Morgan’s laws The negation of a conjunction of two statements is logically equivalent to the disjunction of their negations. The negation of a conjunction of two statements is logically equivalent to the disjunction of their negations. ¬(p  q)  ¬p  ¬q ¬(p  q)  ¬p  ¬q The negation of the disjunction of two statements is logically equivalent to the conjunction of their negation. The negation of the disjunction of two statements is logically equivalent to the conjunction of their negation. ¬(p  q)  ¬p  ¬q ¬(p  q)  ¬p  ¬q

pq ¬p¬p¬p¬p ¬q¬q¬q¬q pqpqpqpq ¬(p  q) ¬p  ¬q De Morgan’s laws

pq ¬p¬p¬p¬p ¬q¬q¬q¬q pqpqpqpq ¬(p  q) ¬p¬q¬p¬q¬p¬q¬p¬q De Morgan’s laws

Tautologies and Contradiction A tautology is a statement form that is always true regardless of the truth values of individual statements substituted for its statement variables. A tautology is a statement form that is always true regardless of the truth values of individual statements substituted for its statement variables. A contradiction is a statement form that is always false regardless of the truth values of individual statements substituted for its statement variables. A contradiction is a statement form that is always false regardless of the truth values of individual statements substituted for its statement variables.

Tautologies and Contradiction p ¬p¬p¬p¬p p¬pp¬pp¬pp¬p p¬pp¬pp¬pp¬p Tautology Contradiction

Logical Equivalences Given any statement variables p, q, and r, a tautology t and a contradiction c the following logical equivalences hold. Given any statement variables p, q, and r, a tautology t and a contradiction c the following logical equivalences hold. 1. Commutative law p  q  q  p p  q  q  p 2. Associative law (p  q)  r  p  (q  r ) (p  q)  r  p  (q  r ) 3. Distributive law p  (q  r)  (p  q)  (p  r ) p  (q  r)  (p  q)  (p  r ) 4. Identity p  t  pp  c  p

Logical Equivalences 5. Negation laws p  ¬p  t p  ¬p  c 6. Double negative law ¬ (¬ p)  p 7. Idempotent pp p pp ppp p pp ppp p pp ppp p pp p 8. Universal bounds laws pt t pc cpt t pc cpt t pc cpt t pc c 9. De Morgan’s laws ¬(p  q)  ¬p  ¬q¬(p  q)  ¬p  ¬q

Logical Equivalences 10. Absorption laws p  (p  q)  p p  (p  q)  p 11. Negations of t and c ¬t  c¬c  t¬t  c¬c  t¬t  c¬c  t¬t  c¬c  t

Simplifying statements Verify the equivalence Verify the equivalence ¬(¬p  q)  (p  q)  p ¬(¬p  q)  (p  q)  p By De Morgan’s By De Morgan’s  (¬(¬p)  ¬q)  (p  q)  (¬(¬p)  ¬q)  (p  q) By double negative law By double negative law  (p  ¬q)  (p  q)  (p  ¬q)  (p  q) By distributive law By distributive law  p  (¬q  q)  p  (¬q  q) By negation law By negation law  p  c  p  c By identity law By identity law  p  p

Simplifying statements Verify the equivalence Verify the equivalence ¬(p  ¬q)  (¬p  ¬ q)  ¬p ¬(p  ¬q)  (¬p  ¬ q)  ¬p

Simplify the following expressions. State the rule you are using at each stage.  ((  p  q)  (  p  q))  (p  q)  ((  p  q)  (  p  q))  (p  q)  (  (  p  q)  (  p  q))  (p  q) De Morgan’s law  (  (  )p  q)  (  (  )p  (  )q))  (p  q)De Morgan’s law  (p  q)  (p  q)  (p  q)Double negative law  p  (  q  q)  (p  q)Distributive laws  (p  t)  (p  q)Negation laws  p  (p  q)Universal bounds laws  pq pq pq pq