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Mathematics for Computing

Mathematics for Computing. Lecture 2: Computer Logic and Truth Tables Dr Andrew Purkiss-Trew Cancer Research UK a.purkiss@mail.cryst.bbk.ac.uk. Logic. Propositions Connective Symbols / Logic gates Truth Tables Logic Laws. Propositions.

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Mathematics for Computing

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  1. Mathematics for Computing Lecture 2: Computer Logic and Truth Tables Dr Andrew Purkiss-Trew Cancer Research UK a.purkiss@mail.cryst.bbk.ac.uk

  2. Logic • Propositions • Connective Symbols / Logic gates • Truth Tables • Logic Laws

  3. Propositions • Definition: A proposition is a statement that is either true or false. Which ever of these (true or false) is the case is called the truth value of the proposition.

  4. Connectives • Compound propositione.g. ‘If Brian and Angela are not both happy, then either Brian is not happy or Angela is not happy’ • Atomic proposition:‘Brian is happy’ ‘Angela is happy’ • Connectives:and, or, not, if-then

  5. Connective Symbols

  6. Conjugation • Logical ‘and’ • Symbol ٨ • Written p٨q • Alternative forms p & q, p . q, pq • Logic gate version p pq q

  7. Disjunction • Logical ‘or’ • Symbol ٧ • Written p ٧ q • Alternative form p + q • Logic gate version p p + q q

  8. Negation • Logical ‘not’ • Symbol ~ • Written ~p • Alternative forms ¬p, p’, p • Logic gate version p ~p

  9. Truth Tables

  10. Compound Propositions ~(p ٨ ~q)

  11. Tautologies • Always true

  12. Contradictions • Always false

  13. Website for Lecture Notes • http://www.cryst.bbk.ac.uk/~bpurk01/MfC/index2007.html

  14. End of First Logic 1? • Place marker

  15. Mathematics for Computing Lecture 3: Computer Logic and Truth Tables 2 Dr Andrew Purkiss-Trew Cancer Research UK a.purkiss@mail.cryst.bbk.ac.uk

  16. Logical Equivalence • Logical ‘equals’ • Symbol ≡ • Written p≡p

  17. Conditional • Logical ‘if-then’ • Symbol → • Written p → q

  18. Biconditional • Logical ‘if and only if’ • Symbol ↔ • Written p ↔ q

  19. converse and contrapositive • The converse of p → q is q → p • The contrapositive of p → q is ~q → ~p

  20. Laws of Logic • Laws of logic allow use to combine connectives and simplify propositions.

  21. Double Negative Law • ~ ~ p ≡p

  22. Implication Law • p → q ≡ ~p ٧ q

  23. Equivalence Law • p ↔ q ≡ (p → q) ٨ (q → p)

  24. Idempotent Laws • p ٨ p ≡p • p ٧ p ≡p

  25. Commutative Laws • p ٨ q ≡q ٨ p • p ٧ q ≡q ٧ p

  26. Associative Laws • p ٨ (q ٨ r) ≡ (p ٨ q) ٨ r • p ٧ (q ٧ r) ≡ (p ٧ q) ٧ r

  27. Distributive Laws • p ٨ (q ٧ r) ≡ (p ٨ q) ٧ (p ٨ r) • p ٧ (q ٨ r) ≡ (p ٧ q) ٨ (p ٧ r)

  28. Identity Laws • p ٨ T ≡p • p ٧ F ≡p

  29. Annihilation Laws • p ٨ F ≡F • p ٧ T ≡T

  30. Inverse Laws • p ٨ ~p ≡F • p ٧ ~p ≡T

  31. Absorption Laws • p ٨ (p ٧ q) ≡p • p ٧ (p ٨ q) ≡p

  32. de Morgan’s Laws • ~(p ٨ q) ≡~p ٧ ~q • ~(p ٧ q) ≡~p ٨ ~q

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