(Boolean) Propositional Logic
A proposition is either True or False
The Logic of
Propositions
Example:
There are 6 regular solids.
5
True
False
Non-examples:
Wake up!
Where am I?
lec 2W.1
February 10, 2010
Albert R Meyer
English to Math
lec 2W.2
February 10, 2010
Albert R Meyer
English to Math
Greeks carry Swords or Javelins
Greeks carry Bronze or Copper swords
True even if a Greek carries both
a Sword and a Javelin
Bronze or Copper but not both
G
(S
J)
lec 2W.3
February 10, 2010
Albert R Meyer
G
(B C)
lec 2W.4
February 10, 2010
Albert R Meyer
Definition of XOR
Definition of OR
The value of (P OR Q) is T iff
P is T, or Q is T, or both are T.
The value of (P XOR Q) is T iff
exactly one of P and Q is T.
Truth Table for XOR
Truth Table for OR
P
P OR Q
P
Q
P XOR Q
T
T
T
T
T
F
T
F
T
T
F
T
F
T
T
F
T
T
F
Albert R Meyer
Q
F
F
F
F
F
February 10, 2010
F iff both
P,Q are F
lec 2W.5
Albert R Meyer
February 10, 2010
lec 2W.6
1
Definition of AND
Definition of NOT
The value of (P AND Q) is T iff
both P and Q are T.
Truth Table for AND
P
T
T
T
F
F
F
T
F
F
F
Truth Table for NOT (P)
P AND Q
T
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Q
The NOT(P) is T iff P is F.
F
February 10, 2010
P
T
Truth Assignments
February 10, 2010
lec 2W.9
Example: Suppose environment, v,
assigns
v(P) = T, v(Q)= T, v(R) = F.
Truth value of
( (P AND Q) ) OR (R XOR ( Q))
FT
FF
FT T T
F
lec 2W.10
February 10, 2010
Albert R Meyer
DeMorgans Law
is equivalent to
Two propositional formulas are
equivalent
iff they have the same truth
value in all environments.
February 10, 2010
lec 2W.8
February 10, 2010
Albert R Meyer
Equivalence
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T
Evaluation in an Environment
A truth assignment assigns a value T
or F to each propositional variable.
Computer scientists call assignment
of values to variables an environment.
If we know the environment, we can
find the value of a propositional
formula.
Albert R Meyer
F
F
lec 2W.7
NOT(P)
lec 2W.11
P
Q
T
T
F
F
T
F
T
F
Albert R Meyer
(P Q)
F
F
F
T
T
F
T
T
F
T
February 10, 2010
F
T
F
F
F
T
F
T
F
T
lec 2W.13
2
Definition of IMPLIES
DeMorgans Law
The value of (P IMPLIES Q) is F iff
P is T and Q is F.
Truth Table for IMPLIES ()
is equivalent to
P
Q
T
T
F
F
T
F
T
F
(P Q)
F
F
F
T
T
F
T
T
F
T
F
T
F
F
F
T
F
P
Albert Meyer
February R 10, 2010
lec 2W.14
A True Implication
February 10, 2010
lec 2W.16
T
T
F
T
February 10, 2010
lec 2W.15
Albert R Meyer
February 10, 2010
lec 2W.17
A True Implication
(1=-1) IMPLIES (I am Pope)
We reasoned correctly to
reach the false conclusion
from the false hypothesis.
February 10, 2010
F
(1=-1) IMPLIES (I am Pope)
We reasoned correctly to
reach the false conclusion
A True Implication
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F
A True Implication
(1=-1) IMPLIES (I am Pope)
We reasoned correctly to
reach the false conclusion
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Albert R Meyer
T
F
Same final column, so equivalent
-- proof by Truth Table
T
F
T
PQ
T
F
Q
T
T
(1=-1) IMPLIES (I am Pope)
We reasoned correctly to
reach the false conclusion
from the false hypothesis.
lec 2W.18
Albert R Meyer
February 10, 2010
lec 2W.19
3
A True Implication
Satisfiability & Validity
(1=-1) IMPLIES (I am Pope)
The whole implication is true,
even though both conclusion
& hypothesis are false.
lec 2W.20
February 10, 2010
Albert R Meyer
Verifying Valid, Satisfiable
A formula is valid iff it is
true in all environments.
Albert R Meyer
February 10, 2010
lec 2W.21
Efficient Test for Satisfiability?
Truth table size doubles with
each additional variable
--exponential growth. Makes
truth tables impossible when
there are hundreds of variables.
(In current digital circuits,
there are millions of variables.)
February 10, 2010
Albert R Meyer
A formula is satisfiable iff it
is true in some environment.
The P=NP? question is equivalent
to asking if there is an efficient
(polynomial rather than
exponential time) procedure
to check satisfiability.
Albert R Meyer
February 10, 2010
Digital Logic
Java Logical Expression
AND
OR
if ((x>0) || (x <= 0 && y>100))
(more code)
half adder
from http://en.wikipedia.org/wiki/Adder_(electronics)
Albert R Meyer
February 10, 2010
lec 2W.38
Albert R Meyer
February 10, 2010
lec 2W.40
4
Digital Logic
A
B
Team Problems
d
cin
cout
s
c
Problems
14
full adder
Albert R Meyer
February 10, 2010
lec 2W.41
Albert R Meyer
February 10, 2010
lec 2W.42
5
MIT OpenCourseWare
http://ocw.mit.edu
6.042J / 18.062J Mathematics for Computer Science
Spring 2010
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