Transcript Mathematics for Computing - Birkbeck, University of London

Mathematics for Computing
Lecture 2:
Computer Logic and Truth Tables
Dr Andrew Purkiss-Trew
Cancer Research UK
[email protected]
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
Connective
Symbol
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
T
T
F
F
q
T
F
T
F
p٨q
T
F
F
F
p
T
T
F
F
p
T
F
~p
F
T
q
T
F
T
F
p٧q
T
T
T
F
Compound Propositions
~(p ٨ ~q)
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
Tautologies
Always true
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-Trew
Cancer Research UK
[email protected]
Logical Equivalence
Logical ‘equals’
Symbol ≡
Written p ≡ p
p
T
T
F
F
q
T
F
T
F
~p
F
F
T
T
~q
F
T
F
T
~p ٨ ~q ~(~p ٨ ~q) p ٧ q
F
T
T
F
T
T
F
T
T
T
F
F
Conditional
Logical ‘if-then’
Symbol →
Written p → q
p
T
T
F
F
q
T
F
T
F
p→q
T
F
T
T
Biconditional
Logical ‘if and only if’
Symbol ↔
Written p ↔ q
p
T
T
F
F
q
T
F
T
F
p↔q
T
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 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
p٨p≡p
p٧p≡p
Commutative Laws
p٨q≡q٨p
p٧q≡q٧p
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
p٨T≡p
p٧F≡p
Annihilation Laws
p٨F≡F
p٧T≡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