Propositional Logic
CS 1050
(Rosen Section 1.1, 1.2)
Proposition
A proposition is a statement that is either
true or false, but not both.
• Atlanta was the site of the 1996
Summer Olympic games.
• 1+1 = 2
• 3+1 = 5
• What will my CS1050 grade be?
Definition 1. Negation of p
Let p be a proposition.
The statement “It is
not the case that p” is
also a proposition,
called the “negation of
p” or ¬p (read “not p”)
p = The sky is blue.
p = It is not the case that
the sky is blue.
p = The sky is not blue.
Table 1.
The Truth Table for the
Negation of a Proposition
p
¬p
T
F
F
T
Definition 2. Conjunction of p
and q
Let p and q be
propositions. The
proposition “p and q,”
denoted by pq is true
when both p and q are
true and is false
otherwise. This is
called the conjunction
of p and q.
Table 2. The Truth Table for
the Conjunction of two
propositions
p
q
pq
T
T
F
F
T
F
T
F
T
F
F
F
Definition 3. Disjunction of p
and q
Table 3. The Truth Table for
the Disjunction of two
propositions
p
q
pq
T
T
F
F
T
F
T
F
T
T
T
F
Let p and q be
propositions. The
proposition “p or q,”
denoted by pq, is the
proposition that is false
when p and q are both
false and true otherwise.
Definition 4. Exclusive or of p
and q
Table 4. The Truth Table for
the Exclusive OR of two
propositions
p
q
pq
T
T
F
F
T
F
T
F
F
T
T
F
Let p and q be
propositions. The
exclusive or of p and q,
denoted by pq, is the
proposition that is true
when exactly one of p
and q is true and is
false otherwise.
Definition 5. Implication pq
Let p and q be propositions.
The implication pq is the
proposition that is false when
p is true and q is false, and
true otherwise. In this
implication p is called the
hypothesis (or antecedent or
premise) and q is called the
conclusion (or
consequence).
Table 5. The Truth Table for
the Implication of pq.
p
q
pq
T
T
F
F
T
F
T
F
T
F
T
T
Implications
•
•
•
•
•
•
•
•
If p, then q
p implies q
if p,q
p only if q
p is sufficient for q
q if p
q whenever p
q is necessary for p
• Not the same as the
if-then construct
used in
programming
languages such as
If p then S
Implications
How can both p and q be false, and pq be true?
•Think of p as a “contract” and q as its “obligation” that is
only carried out if the contract is valid.
•Example: “If you make more than $25,000, then you must
file a tax return.” This says nothing about someone who
makes less than $25,000. So the implication is true no
matter what someone making less than $25,000 does.
•Another example:
p: Bill Gates is poor.
q: Pigs can fly.
pq is always true because Bill Gates is not poor. Another
way of saying the implication is
“Pigs can fly whenever Bill Gates is poor” which is true
since neither p nor q is true.
Related Implications
Converse of
pq
is
qp
Contrapositive
of p  q
is the proposition
q  p
Definition 6. Biconditional
Table 6. The Truth Table for
the biconditional pq.
p
q
pq
T
T
F
F
T
F
T
F
T
F
F
T
Let p and q be
propositions. The
biconditional pq is the
proposition that is true
when p and q have the
same truth values and is
false otherwise. “p if and
only if q, p is necessary
and sufficient for q”
Practice
p: You learn the simple things well.
q: The difficult things become easy.
• You do not learn the
simple things well. p
• If you learn the simple
things well then the
difficult things become
easy.
pq
• If you do not learn the
simple things well, then
the difficult things will
not become easy.
p  q
• The difficult things
become easy but you
did not learn the simple
things well. q  p
• You learn the simple
things well but the
difficult things did not
become easy.
p  q
Truth Table Puzzle
Steve would like to determine the relative
salaries of three coworkers using two facts
(all salaries are distinct):
• If Fred is not the highest paid of the three,
then Janice is.
• If Janice is not the lowest paid, then Maggie
is paid the most.
Who is paid the most and who is paid the least?
p : Janice is paid the most.
q: Maggie is paid the most.
r: Fred is paid the most.
s: Janice is paid the least.
p
T
F
F
F
F
q
F
T
F
T
F
r
F
F
T
F
T
s
F
T
T
F
F
Fred, Maggie, Janice
rp
T
F
T
F
T
•If Fred is not the highest paid
of the three, then Janice is.
•If Janice is not the lowest paid,
then Maggie is paid the most.
s q (rp) (sq)
F
F
T
F
T
T
T
F
F
F
p : Janice is paid the most.
q: Maggie is paid the most.
r: Fred is paid the most.
s: Janice is paid the least.
p
T
F
F
F
F
q
F
T
F
T
F
r
F
F
T
F
T
s
F
T
T
F
F
rp
T
F
T
F
T
•If Fred is not the highest paid
of the three, then Janice is.
•If Janice is the lowest paid,
then Maggie is paid the most.
s q
T
T
F
T
T
(rp) (sq)
T
F
F
F
T
Fred, Janice, Maggie or Janice, Maggie, Fred
or Janice, Fred, Maggie
Bit Operations
A computer bit has two possible values: 0 (false) and 1
(true). A variable is called a Boolean variable is its value is
either true or false.
Bit operations correspond to the logical connectives:
 OR
 AND
 XOR
Information can be represented by bit strings, which are
sequences of zeros and ones, and manipulated by
operations on the bit strings.
Truth tables for the bit
operations OR, AND, and XOR

0
0 1
0
0 1
1 1
1
1 0

0
0
1
1

0
0
0 0
1
0 1
1
1
Logical Equivalence
• An important technique in proofs is to
replace a statement with another
statement that is “logically equivalent.”
• Tautology: compound proposition that is
always true regardless of the truth values
of the propositions in it.
• Contradiction: Compound proposition
that is always false regardless of the truth
values of the propositions in it.
Logically Equivalent
• Compound propositions P and Q are
logically equivalent if PQ is a
tautology. In other words, P and Q have
the same truth values for all
combinations of truth values of simple
propositions.
• This is denoted: PQ (or by P  Q)
Example: DeMorgans
• Prove that (pq)  (p  q)
(pq) p q
(p  q)
p q
(pq)
TT
TF
FT
FF
T
F
F
F
F
T
F
F
T
F
T
F
F
T
T
T
F
T
F
T
Illustration of De Morgan’s Law
(pq)
p
q
Illustration of De Morgan’s Law
p
p
Illustration of De Morgan’s Law
q
q
Illustration of De Morgan’s Law
p  q
p
q
Example: Distribution
Prove that: p  (q  r)  (p  q)  (p  r)
p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r
T
F
T
F
T
F
T
F
qr p(qr) pq pr
T
T
T
T
F
T
T
T
F
T
T
T
F
T
T
T
T
T
T
T
F
F
T
F
F
F
F
T
F
F
F
F
(pq)(pr)
T
T
T
T
T
F
F
F
Prove: pq(pq)  (qp)
pq
TT
TF
FT
FF
pq
T
F
F
T
pq qp
T
T
F
T
T
F
T
T
(pq)(qp)
T
F
F
T
We call this biconditional equivalence.
List of Logical Equivalences
pT  p;
pF  p
Identity Laws
pT  T;
pF  F
Domination Laws
pp  p;
pp  p
Idempotent Laws
(p)  p
Double Negation Law
pq  qp; pq  qp
Commutative Laws
(pq) r  p (qr); (pq)  r  p  (qr)
Associative Laws
List of Equivalences
p(qr)  (pq)(pr)
p(qr)  (pq)(pr)
Distribution Laws
(pq)(p  q)
(pq)(p  q)
De Morgan’s Laws
p  p  T
p  p  F
(pq)  (p  q)
Miscellaneous
Or Tautology
And Contradiction
Implication Equivalence
pq(pq)  (qp)
Biconditional Equivalence