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Mathematics for Computing

This lecture by Dr. Andrew Purkiss-Trew introduces the fundamental concepts of computer logic, including propositions, connectives, and truth tables. It defines propositions as statements with truth values (true or false) and explores the various logical connectives such as 'and', 'or', and 'not', along with their symbolic representations. The lecture further discusses compound propositions, equivalence, implications, and essential laws of logic guiding the simplification and proof of logical statements. For detailed notes, visit the provided website.

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Mathematics for Computing

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Presentation Transcript

  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 us to combine connectives and simplify propositions and prove that logical equivalences are correct.

  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

  33. End of Logic

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