Mathematics for Computing

Lecture 2:Computer Logic and Truth Tables

Dr Andrew Purkiss-TrewCancer Research UK

a.purkiss@mail.cryst.bbk.ac.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 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

Connective Symbols

Connective Symbol

and ٨

or ٧

not ~ or ¬

if-then →

if-and-only-if ↔

Conjugation

Logical ‘and’

Symbol ٨Written p ٨ q Alternative forms p & q, p . q, pqLogic gate version

pq pq

Disjunction

Logical ‘or’

Symbol ٧Written p ٧ qAlternative form p + qLogic gate version

pq p + q

Negation

Logical ‘not’

Symbol ~Written ~pAlternative forms ¬p, p’, p Logic gate version

p ~p

Truth Tables

p ~p

T F

F T

p q p ٨ q

T T T

T F F

F T F

F F F

p q p ٧ q

T T T

T F T

F T T

F F F

Compound Propositions

p q ~q

T T F

T F T

F T F

F F T

~(p ٨ ~q)

p q ~q p ٨~q

T T F F

T F T T

F T F F

F F T F

p q ~q p ٨~q ~(p ٨ ~q)

T T F F T

T F T T F

F T F F T

F F T F T

p q

T T

T F

F T

F F

Tautologies

Always true

p ~p p ٧ ~p

T F T

F T T

p ~p p ٧ ~p

T F T

F T T

Contradictions

Always false

p ~p p ٨ ~p

T F F

F T F

Website for Lecture Notes

http://www.cryst.bbk.ac.uk/~bpurk01/MfC/index2007.html

End of First Logic 1?

Place marker

Mathematics for Computing

Lecture 3:Computer Logic and Truth Tables 2

Dr Andrew Purkiss-TrewCancer Research UK

a.purkiss@mail.cryst.bbk.ac.uk

Logical Equivalence

Logical ‘equals’

Symbol ≡

Written p ≡ p

p q ~p ~q ~p ٨ ~q

~(~p ٨ ~q)

T T F F F T

T F F T F T

F T T F F T

F F T T T F

p ٧ q

T

T

T

F

Conditional

Logical ‘if-then’

Symbol →Written p → q

p q p → q

T T T

T F F

F T T

F F T

Biconditional

Logical ‘if and only if’

Symbol ↔Written p ↔ q

p q p ↔ q

T T T

T F F

F T F

F F T

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

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

End of Logic