Proposition
A proposition is a statement that is either
true or false, but not both.
•  Clemson will defeat Georgia in football
this fall.
•  1+1 = 2
•  3+1 = 5
•  What will be my grade in CPSC 2070?
Propositional Logic
CPSC 2070 Discrete Structures
Rosen (6th Ed.) 1.1, 1.2
Definition 1. Negation of p
LOGICAL
We can define operations on propositions!
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.
Table 1.
The Truth Table for the
Negation of a Proposition
p
¬p
T
F
F
T
¬p = The sky is not blue.
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.
1
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).
Politician Example
Implications
•
•
•
•
•
•
•
If p, then q
p implies q
if p,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
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”
p
q
p→q
T
T
F
F
T
F
T
F
T
F
T
T
Related Implications
Converse
of p → q
is the proposition
q→p
Contrapositive
of p → q
is the proposition
¬q → ¬p
Inverse
of p → q
is the proposition
¬p → ¬q
Definition 6. Biconditional
Table 6. The Truth Table for
the biconditional p↔q.
Table 5. The Truth Table for
the Implication of p→q.
Compound Propositions
We can also combine operations to create
compound propositions such as:
Don’t try to do it
Test all possible
¬(p ∨ q) ∧ q
all in your head!
combinations of T/
F
p
q
(p ∨ q)
¬(p ∨ q)
¬(p ∨ q) ∧ q
T
T
T
F
F
T
F
T
F
F
F
T
T
F
F
F
F
F
T
F
2
Special Types of Compound
Propositions
•  Contradiction: Compound proposition
that is always false regardless of the
truth values of the propositions in it.
p ∧ ¬p is a contradiction
•  Tautology: compound proposition that
is always true regardless of the truth
values of the propositions in it.
p ∨ ¬p is a tautology
Practice with English Sentences
p: You learn the simple things well.
q: The difficult things become easy.
•  The difficult things
become easy but you
did not learn the simple
things well.
•  You learn the simple
things well but the
difficult things did not
become easy.
•  You do not learn the
simple things well.
•  If you learn the simple
things well then the
difficult things become
easy.
•  If you do not learn the
simple things well, then
the difficult things will
not become easy.
Practice with English Sentences
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.
•  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.
Some Applications of
Propositional Logic
p ∧ ¬q
¬p → ¬q
Truth Table Puzzle
Steve would like to determine the relative
salaries of three coworkers using two
facts:
•  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
¬r→p
T
F
T
F
T
If Janice is not the lowest
If Fred is not the
paid, then Maggie is paid
highest paid of the
the most.
three, then Janice is.
¬s →q (¬r→p)∧ (¬s→q)
F
F
T
F
T
T
T
F
F
F
Fred, Maggie, Janice
3
Knights and Knaves Puzzles
Let a be the statement that “A is a Knight” and b be the statement that
“B is a Knight.”
A says “B and I are both Knights.” If A is a Knight then a is true and the
statement (a∧b) is true. If A is a Knave, then a is false and the
statement is false. These are the two cases that are “true” in that
they are possible solutions to the puzzle. We can code this with the
biconditional operator ↔. (Recall that p↔q is true when both
propositions are true or both propositions are false.)
On a remote island there live Knights and
Knaves. Knights always tell the truth
and Knaves always lie.
B says “A is a Knave.” Using the same logic add b ↔ ¬a to the table.
You meet two people on the island—A
and B. A says: “B and I are both
Knights.” B says: “A is a Knave.”
Determine, if possible, which group A and
B belong to.
A says “At least one of us is a Knave” and B
says nothing.
a (A is a
Knight)
b (B is a
Knight)
¬a ∨ ¬b
a ↔ (¬a ∨ ¬b)
T
T
F
F
T
F
T
F
F
T
T
T
F
T
F
F
a (A is a Knight)
b (B is a Knight)
(a∧b)
a ↔ (a∧b)
b ↔ ¬a
T
T
T
T
F
T
F
F
F
T
F
T
F
T
T
F
F
F
T
F
The Hat Puzzle
Three students who made an “A” in CPSC 2070 are told to
stand in a straight line, one in front of the other. A hat is put on
each of their heads. They are told that each of these hats was
selected from a group of five hats: two black hats and three white
hats. The first student, standing at the front of the line, can’t see
either of the students behind her or their hats. The second
student, in the middle, can see only the first student and her hat.
The last student, at the rear, can see both other students and
their hats.
None of them can see the hat on their own head. They are asked
to deduce its color. The last student in line is asked if he knows
the color of his hat and says he cannot be sure. The second
student in line is asked if he knows the color of his hat and says
he cannot be sure. The student at the front of the line then says:
“My hat is white.”
She is correct. How did she come to this conclusion?
Bit Operations
Last
Second
First
W
W
W
W
W
B
W
W
B
B
W
B
B
B
W
W
W
B
B
B
W
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 zeroes and ones, and manipulated by
operations on the bit strings.
4
Truth tables for the bit
operations OR, AND, and XOR
∨
0
0
0 1
1
1 1
1
∧
0
0
0 0
1
0 1
⊕
0
0
0 1
1
1 0
1
1
Binary Math Review
Decimal has digits 0-9
0
0
1
3
___
___
___
___
3
2
1
10 10 10 100
0+1 = 1
1+0 = 1
1+1 = 10
10 + 1 = 11
11 + 1 = 100
Binary has digits 0-1
1
1
0
1
___
___
___
___
3
2
1
2
2
2
20
Program to add 3 binary digits
p
q
r
Output
1
Output
2
0
0
0
0
0
0
0
1
0
1
0
1
0
0
1
0
1
1
1
0
1
0
0
0
1
1
0
1
1
0
1
1
0
1
0
1
1
1
1
Output 1 = (p∧q) ∨ (p∧r) ∨ (q∧r)
1
Output 2 = p ⊕ q ⊕ r
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
Full Adder: Computers contain switches (logic gates) corresponding
to our logic operations
Example: DeMorgans
•  Prove that ¬(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
¬(p∨q) ¬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
T
T
T
F
T
T
F
T
T
F
T
T
T
T
T
F
F
T
F
F
F
F
F
F
p∨r
T
T
T
T
T
F
T
F
(p∨q)∧(p∨r)
T
T
T
T
T
F
F
F
5
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
Table 6 in Section 1.2
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)
Distributive Laws
¬(p∨q)⇔(¬p ∧ ¬q)
¬(p∧q)⇔(¬p ∨ ¬q)
De Morgan’s Laws
p ∨ (p∧q) ⇔ p
p ∧ (p∨q) ⇔ p
Absorption Laws
p ∨ ¬p ⇔ T
p ∧ ¬p ⇔ F
Negation Laws
(p→q) ⇔ (¬p ∨ q)
Implication Equivalence
Or Tautology; And Contradiction
6