Mathematics for Computing Lecture 2: Computer Logic and Truth Tables Dr Andrew Purkiss-Trew Cancer Research UK

Logic Propositions Connective Symbols / Logic gates Truth Tables Logic Laws

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.

Connectives Compound proposition e.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

Connective Symbols ConnectiveSymbol and ٨ or ٧ not ~ or ¬ if-then → if-and-only-if ↔

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

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

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

Truth Tables p~p TF FT pqp ٨ qp ٨ q TTT TFF FTF FFF pqp ٧ qp ٧ q TTT TFT FTT FFF

Compound Propositions pq~q TTF TFT FTF FFT ~(p ٨ ~q) pq~qp ٨~q TTFF TFTT FTFF FFTF pq~qp ٨~q ~(p ٨ ~q) TTFFT TFTTF FTFFT FFTFT pq TT TF FT FF

Tautologies Always true p~pp ٧ ~p TFT FTT p~pp ٧ ~p TFT FTT

Contradictions Always false p~pp ٨ ~p TFF FTF

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Mathematics for Computing Lecture 3: Computer Logic and Truth Tables 2 Dr Andrew Purkiss-Trew Cancer Research UK

Logical Equivalence Logical ‘equals’ Symbol ≡ Written p ≡ p pq~p~q~p ٨ ~q~(~p ٨ ~q) TTFFFT TFFTFT FTTFFT FFTTTF p ٧ q T T T F

Conditional Logical ‘if-then’ Symbol → Written p → q pqp → q TTT TFF FTT FFT

Biconditional Logical ‘if and only if’ Symbol ↔ Written p ↔ q pqp ↔ q TTT TFF FTF FFT

converse and contrapositive The converse of p → q is q → p The contrapositive of p → q is ~q → ~p

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

Double Negative Law ~ ~ p ≡ p

Implication Law p → q ≡ ~ p ٧ q

Equivalence Law p ↔ q ≡ (p → q) ٨ ( q → p)

Idempotent Laws p ٨ p ≡ p p ٧ p ≡ p

Commutative Laws p ٨ q ≡ q ٨ p p ٧ q ≡ q ٧ p

Associative Laws p ٨ (q ٨ r) ≡ ( p ٨ q) ٨ r p ٧ (q ٧ r) ≡ ( p ٧ q) ٧ r

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)

Identity Laws p ٨ T ≡ p p ٧ F ≡ p

Annihilation Laws p ٨ F ≡ F p ٧ T ≡ T

Inverse Laws p ٨ ~p ≡ F p ٧ ~p ≡ T

Absorption Laws p ٨ (p ٧ q) ≡ p p ٧ (p ٨ q) ≡ p

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