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Symbolic Logic

Symbolic Logic. 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.

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Symbolic Logic

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

  2. Symbolic Logic • The main purpose of logic is to buildthe thinking methods. • Provide rules,techniques, for makingdecision in an argument,validating a deduction. • Inclassical logic, only phrases , assertions with one truthvalue are allowed: TRUE or FALSE, without ambiguity • Such boolean assertions are called propositions.

  3. Symbolic Logic • A proposition is a statement that is either true orfalse. • The purpose of propositionallogic is to provide complex construction of rules, from anonymouspropositions calledpropositional variables • If p is a proposition ,the negation of p, denoted by ¬p, is a proposition which means " itis false that". Then if p is true , ¬p is false, andif p is false , ¬p is true.

  4. Symbolic Logic • A propositionconsisting of only a single propositional variable or singleconstant (true or false) is called an atomic proposition. • All nonatomic propositions are called compoundpropositions. All compound propositions contain at least onelogical connective.

  5. Symbolic Logic • If p and q are propositions, the conjunction of pand q is the proposition " p and q” denoted by pq. • The proposition pq is true if p and q areboth true, and false otherwise; this is describe by the followingtruth table:

  6. Symbolic Logic • The disjunction of pand q is the proposition " p or q” denoted by pq. • The proposition pq is true if at least one ofthe two propositions p and p is true, and false when p and q areboth false; this is described by thefollowing truth table:

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

  8. (pq )¬r

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

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

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

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

  13. De Morgan’s laws

  14. De Morgan’s laws

  15. 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 contradiction is a statement form that is always false regardless of the truth values of individual statements substituted for its statement variables.

  16. Tautologies and Contradiction Tautology Contradiction

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

  18. Logical Equivalences • Negation laws p¬p t p ¬p c • Double negative law ¬ (¬ p)p • Idempotent pp p pp p • Universal bounds laws pt t pc c • De Morgan’s laws ¬(pq)  ¬p ¬q ¬(p  q)  ¬p ¬q

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

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

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

  22. 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) 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

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