Download Section 1

Section 1.3 Truthtellers, liars and propositional logic
We call a sentence a proposition
p:
q:
r:
A formal proposition
1.
2. Given propositions p and q, their conjunction, written p  q is
3. Given propositions p and q, their disjunction, written p V q
4. Given a proposition p, the compound statement p
Truth Tables
bit operations corresponding to logical connectives
1010 1100
0011 1011
AND
OR
XOR
Combining operations
Two statements are logically equivalent
Proposition 1 (p. 32) (DeMorgan’s Laws) Let p and q be any
propositions. Then
1. (p V q) is logically equivalent to p  q
2. (p  q) is logically equivalent to p V q
Example
A tautology is
A contradiction is
Table of Equivalences
Commutative
Associative
Distributive
Identity
Negation
Double negative
Idempotent
DeMorgan's Laws
Universal
Absorption
pqqp
(p  q)  r = p  (q  r)
p  (q V r)  (p  q) V (p  r)
pTp
p V p  T
 
( p)  p
ppp

(p  q)  p V q
pVTT
p  (p V q)  p
pVqqVp
(p V q) V r  p V (q V r)
p V (q  r)  (p V q)  (p V r)
pVFp
p  p  F
pVpp

(p V q)  p  q
pFF
p V (p  q)  p
Proof Example
Show (p  q) V q  p V q
Section 1.5 Implications
An implication is
Truth table for the implication
p
q
T
T
T
F
F
T
F
F
pq
The biconditional statement:
Notation: p  q.
p
q
T
T
T
F
F
T
F
F
pq
Terminology and Precedence
Alternate Ways of Expressing Implications
English Word
Logical
Connective
Logical
Expression
and; but; also; moreover
conjunction
pq
or
disjunction
pVq
implication
pq
if p, then q
p implies q
p, therefore q
p only if q
q follows from p
p is a sufficient condition for q
q is a necessary condition for p
p if and only if q
equivalence
pq
p is necessary and sufficient for (biconditional)
q
not p
it is false that p …
it is not true that p …
negation

p
Examples:
The flowers bloom only if you water them.
If it rains hard, it’s necessary to carry and umbrella.
Having one even factor is sufficient to have an even product.
Contrapositives, converses, and inverses
Definition Consider the implication p  q
1. The converse of the implication is
2. The inverse of the implication is
3. The contrapositive of the implication is
Proposition 3
1. An implication and its contrapositive are logically equivalent
2. The converse and inverse of an implication are logically
equivalent
3. The implication is not logically equivalent to its converse
Showing p  q and q  p are logically equivalent and that p  q
and q  p are not
p
q
q
p
pq
q  p
qp
Using Equivalence in Proofs
Statement: If n2 is odd then n is odd
Contrapositive:
Indirect (contrapositive) proof:
Direct proof:
Statement: If a and b are odd numbers, then a•b is odd.
Inverse:
Translating English sentences
If the file is not damaged and the processor is fast, then the
printer is slow.
Translate: The program is efficient only if it executes fast and
doesn’t have a bug.
p:
q:
r:
System Specifications
Consistency
Consistency Example
1. The system is in multiuser state iff it is operating normally.
2. If the system is operating normally, the kernel is functioning.
3. The kernel is not functioning or the system is in interrupt mode.
4. If the system is not in multiuser state then it is in interrupt mode.
5. The system is not in interrupt mode.
p:
q:
r:
s: