Boolean Logic
CS 202, Spring 2007
Epp, sections 1.1 and 1.2
Aaron Bloomfield
1
Administratrivia
• HW 1: due next Friday
– Section 1.1 # 49, 52
– Section 1.2 # 34, 44, 46
• Today’s lecture will be somewhat of a
review
• Next week we will see applications of logic
2
Applications of Boolean logic
•
•
•
•
Computer programs
And computer addition
Logic problems
Sudoku
3
Boolean propositions
• A proposition is a statement that can be
either true or false
– “The sky is blue”
– “I is a Engineering major”
– “x == y”
• Not propositions:
– “Are you Bob?”
– “x := 7”
4
Boolean variables
• We use Boolean variables to refer to
propositions
– Usually are lower case letters starting with p
(i.e. p, q, r, s, etc.)
– A Boolean variable can have one of two
values true (T) or false (F)
• A proposition can be…
– A single variable: p
– An operation of multiple variables: p(qr)
5
Introduction to Logical Operators
• About a dozen logical operators
– Similar to algebraic operators + * - /
• In the following examples,
– p = “Today is Friday”
– q = “Today is my birthday”
6
Logical operators: Not
• A not operation switches (negates) the truth value
• Symbol:  or ~
• In C++ and Java,
the operand is !
p
T
F
p
F
T
• p = “Today is not Friday”
7
Logical operators: And
• An and operation is true if both operands are true
• Symbol: 
– It’s like the ‘A’ in And
• In C++ and Java,
the operand is &&
• pq = “Today is Friday and
today is my birthday”
p
T
T
F
F
q
T
F
T
F
pq
T
F
F
F
8
Logical operators: Or
• An or operation is true if either operands are true
• Symbol: 
• In C++ and Java,
the operand is ||
• pq = “Today is Friday or
today is my birthday (or
p
T
T
F
F
q
T
F
T
F
pq
T
T
T
F
possibly both)”
9
Logical operators: Exclusive Or
• An exclusive or operation is true if one of the operands
are true, but false if both are true
• Symbol: 
• Often called XOR
• pq  (p  q)  ¬(p  q)
• In Java, the operand is ^
(but not in C++)
• pq = “Today is Friday or today
is my birthday, but not both”
p
T
T
F
F
q
T
F
T
F
pq
F
T
T
F
10
Inclusive Or versus Exclusive Or
• Do these sentences mean inclusive or
exclusive or?
– Experience with C++ or Java is required
– Lunch includes soup or salad
– To enter the country, you need a passport or a
driver’s license
– Publish or perish
11
Logical operators: Nand and Nor
• The negation of And and Or, respectively
• Symbols: | and ↓, respectively
– Nand: p|q  ¬(pq)
– Nor: p↓q  ¬(pq)
p
T
T
q
T
F
F
F
T
F
pq pq
T
T
F
T
F
F
T
F
p|q
F
T
pq
F
F
T
T
F
T
12
Logical operators: Conditional 1
• A conditional means “if p then q”
• Symbol: 
p
• pq = “If today is
T
Friday, then today
T
is my birthday”
F
• p→q  ¬pq
F
the
antecedent
q
T
F
T
F
pq ¬pq
T
T
F
F
T
T
T
T
the
consequence
13
Logical operators: Conditional 2
• Let p = “I am elected” and q = “I will lower taxes”
• I state: p  q = “If I
am elected, then I
will lower taxes”
• Consider all
possibilities
p
T
T
F
F
q
T
F
T
F
pq
T
F
T
T
• Note that if p is false, then
the conditional is true regardless of whether q is
true or false
14
Logical operators: Conditional 3
p
T
q
T
T
F
F
F
T
F
p q
F
F
F
T
T
T
F
T
Conditional
Inverse
Converse
Contrapositive
pq
T
pq
T
qp
T
qp
T
F
T
T
T
F
T
T
F
T
F
T
T
15
Logical operators: Conditional 4
• Alternate ways of stating a conditional:
– p implies q
– If p, q
– p is sufficient for q
– q if p
– q whenever p
– q is necessary for p
– p only if q
I don’t like this one
16
Logical operators: Bi-conditional 1
• A bi-conditional means “p if and only if q”
• Symbol: 
p
q
• Alternatively, it means
T
T
“(if p then q) and
T
F
(if q then p)”
F
T
• pq  pq  qp
F
F
• Note that a bi-conditional
has the opposite truth values
of the exclusive or
pq
T
F
F
T
17
Logical operators: Bi-conditional 2
• Let p = “You take this class” and q = “You get a
grade”
p
q pq
• Then pq means
T
T
T
“You take this class if
and only if you get a
T
F
F
grade”
F
T
F
• Alternatively, it means “If
F
F
T
you take this class, then
you get a grade and if you get a grade then you
take (took) this class”
18
Boolean operators summary
not
not
and
or
pq pq
xor
nand
nor
conditional
biconditional
pq
p|q
pq
pq
pq
p
q
p
q
T
T
F
F
T
T
F
F
F
T
T
T
F
F
T
F
T
T
T
F
F
F
F
T
T
F
F
T
T
T
F
T
F
F
F
T
T
F
F
F
T
T
T
T
• Learn what they
memorize the table!
mean,
don’t
just
19
Precedence of operators
• Just as in
precedence
algebra,
operators
have
– 4+3*2 = 4+(3*2), not (4+3)*2
• Precedence order (from highest to lowest):
¬→↔
– The first three are the most important
• This means that p  q  ¬r → s ↔ t
yields: (p  (q  (¬r))) ↔ (s → t)
• Not is always performed before any other
operation
20
Translating English Sentences
• Problem:
– p = “It is below freezing”
– q = “It is snowing”
pq
It is below freezing and it is snowing
p¬q
It is below freezing but not snowing
¬p¬q
It is not below freezing and it is not snowing
It is either snowing or below freezing (or both) pq
p→q
If it is below freezing, it is also snowing
It is either below freezing or it is snowing,(pq)(p→¬q)
but it is not snowing if it is below freezing
• That it is below freezing is necessary and p↔q
sufficient for it to be snowing
•
•
•
•
•
•
21
Translation Example 1
• Heard on the radio:
– A study showed that there was a correlation
between the more children ate dinners with
their families and lower rate of substance
abuse by those children
– Announcer conclusions:
• If children eat more meals with their family, they
will have lower substance abuse
• If they have a higher substance abuse rate, then
they did not eat more meals with their family
22
Translation Example 1
• Let p = “Child eats more meals with family”
• Let q = “Child has less substance abuse
• Announcer conclusions:
– If children eat more meals with their family, they will
have lower substance abuse
• pq
– If they have a higher substance abuse rate, then they
did not eat more meals with their family
• q  p
• Note that p  q and q  p are logically
equivalent
23
Translation Example 1
• Let p = “Child eats more meals with family”
• Let q = “Child has less substance abuse”
• Remember that the study showed a
correlation, not a causation
p
T
T
F
F
q
T
F
T
F
result
T
?
?
T
conclusion
T
F
T
T
24
Translation Example 2
• “I have neither given nor received help on this
exam”
– Rephrased: “I have not given nor received …”
– Let p = “I have given help on this exam”
– Let q = “I have received help on this exam”
• Translation is: pq
p
T
q
T
p
F
pq
F
T
F
F
F
T
F
F
T
T
T
F
F
25
Translation Example 2
• What they mean is “I have not given and I have
not received help on this exam”
– Or “I have not (given nor received) help on this exam”
p
T
q
T
pq
F
(pq)
F
T
F
F
F
T
F
F
F
T
F
F
T
• The problem:  has a higher precedence than 
in Boolean logic, but not always in English
• Also, “neither” is vague
26
Boolean Searches
(101 OR 202) AND bloomfield AND
“computer science”
• Note that Google requires you to capitalize
Boolean operators
• Google defaults to AND; many others do
not
– So the AND’s were optional
• XOR doesn’t work…
28
Bit Operations 1
• Boolean values can be represented as 1 (true)
and 0 (false)
• A bit string is a series of Boolean values
– 10110100 is eight Boolean values in one string
• We can then do operations on these Boolean
strings
– Each column is its own
01011010
Boolean operation
10110100
11101110
29
Bit Operations 2
• Evaluate the following
11000  (01011  11011)
= 11000  (11011)
= 11000
01011
11011
11011
11000
11011
11000
30
&& vs. & in C/C++
Consider the
following:
In C/C++, any value other than 0 is true
int p = 11;
int q = 20;
if ( p && q ) {
}
if ( p & q ) {
}
The binary for the integer 20 is 10100
Notice the double ampersand – this is a
Boolean operation
As p and q are both true, this is true
Notice the single ampersand – this is a
bitwise operation
The binary for the integer 11 is 01011
01011
Bitwise Boolean
10100
And operation:
00000
This evaluates to
zero (false)!
31
&& vs. & in C/C++
• Note that Java does not have this “feature”
– If p and q are int:
• p & q is bitwise
• p && q will not compile
– If p and q are boolean:
• Both p & q and p && q will be a Boolean
operation
• The same holds true for the or operators
(| and ||) in both Java and C/C++
32
Tautology and Contradiction
• A tautology is a statement that is always
true
– p  ¬p will always be true
(Negation Law)
• A contradiction is a statement that is
always false
– p  ¬p will always be false
p
T
F
p  ¬p
T
T
(Negation Law)
p  ¬p
F
F
33
Logical Equivalence
• A logical equivalence means that the two
sides always have the same truth values
– Symbol is ≡ or 
• We’ll use ≡, so as not to confuse it with the biconditional
34
Logical Equivalences of And
• pT≡p
p
T
F
Identity law
T
T
T
• pF≡F
p
T
F
pT
T
F
Domination law
F
F
F
pF
F
F
35
Logical Equivalences of And
• pp≡p
Idempotent law
p
T
F
p
T
F
• pq≡qp
pp
T
F
Commutative law
p
T
q
T
pq
T
qp
T
T
F
F
F
T
F
F
F
F
F
F
F
36
Logical Equivalences of And
• (p  q)  r ≡ p  (q  r)
p q
r
Associative law
pq
(pq)r
qr
p(qr)
T T T
T
T
T
T
T T F
T
F
F
F
T F T
F
F
F
F
T F F
F
F
F
F
F T T
F
F
T
F
F T F
F
F
F
F
F F T
F
F
F
F
F F F
F
F
F
F
37
Logical Equivalences of Or
•
•
•
•
•
pT≡T
pF≡p
pp≡p
pq≡qp
(p  q)  r ≡ p  (q  r)
Identity law
Domination law
Idempotent law
Commutative law
Associative law
38
Corollary of the Associative Law
•
•
•
•
(p  q)  r ≡ p  q  r
(p  q)  r ≡ p  q  r
Similar to (3+4)+5 = 3+4+5
Only works if ALL the operators are the same!
39
Logical Equivalences of Not
• ¬(¬p) ≡ p
• p  ¬p ≡ T
• p  ¬p ≡ F
Double negation law
Negation law
Negation law
40
DeMorgan’s Law
• Probably the most important logical equivalence
• To negate pq (or pq), you “flip” the sign, and
negate BOTH p and q
– Thus, ¬(p  q) ≡ ¬p  ¬q
– Thus, ¬(p  q) ≡ ¬p  ¬q
pq
TT
TF
FT
FF
p
F
F
T
T
q
F
T
F
T
pq (pq) pq pq (pq) pq
T
F
F
T
F
F
F
T
T
T
F
F
F
T
T
T
F
F
F
T
T
F
T
T
41
Yet more equivalences
• Distributive:
p  (q  r) ≡ (p  q)  (p  r)
p  (q  r) ≡ (p  q)  (p  r)
• Absorption
p  (p  q) ≡ p
p  (p  q) ≡ p
42
How to prove two propositions
are equivalent?
• Two methods:
– Using truth tables
• Not good for long formulae
• In this course, only allowed if specifically stated!
– Using the logical equivalences
• The preferred method
• Example: show that:
( p  r )  (q  r )  ( p  q)  r
43
Using Truth Tables
( p  r )  (q  r )  ( p  q)  r
p q r p→r q →r (p→r)(q →r) pq (pq) →r
T T T
T
T
T
T
T
T T F
F
F
F
T
F
T F T
T
T
T
F
T
T F F
F
T
T
F
T
F T T
T
T
T
F
T
F T F
T
F
T
F
T
F F T
T
T
T
F
T
F F F
T
T
T
F
T
44
Using Logical Equivalences
( p  r )  (q  r )  ( p  q)  r
Original statement
(p Definition
r )  (qofimplication
r )  ( p pq)
 qr  p  q
(p DeMorgan’s
r )  (q  Law
r )  ( p( pqq))  
r p  q
Associativity
p  r  q ofrOr (pprq) (rq  r )  p  r  q  r
Re-arranging
p  q  r  r  p  q  r
Idempotent
p  q Law
r 
p
r
r rq  r
45
Logical Thinking
• At a trial:
– Bill says: “Sue is guilty and Fred is innocent.”
– Sue says: “If Bill is guilty, then so is Fred.”
– Fred says: “I am innocent, but at least one of the
others is guilty.”
• Let b = Bill is innocent, f = Fred is innocent, and
s = Sue is innocent
• Statements are:
– ¬s  f
– ¬b → ¬f
– f  (¬b  ¬s)
46
Can all of their statements be true?
• Show: (¬s  f)  (¬b → ¬f)  (f  (¬b  ¬s))
b
f
s ¬b ¬f ¬s ¬sf
¬b→¬f
¬b¬s
f(¬b¬s)
T T T
F
F
F
F
T
F
F
T T F
F
F
T
T
T
T
T
T F T
F
T
F
F
T
F
F
T F F
F
T
T
F
T
T
F
F T T
T
F
F
F
F
T
T
F T F
T
F
T
T
F
T
T
F F T
T
T
F
F
T
T
F
F F F
T
T
T
F
T
T
F
47
Are all of their statements true?
Show values for s, b, and f such
that the equation is true
(s  f )  (b  f )  ( f  (b  s))  T
(s  f )  (b  f )  ( f  (b  s))  T
s  f  (b  f )  f  (b  s)  T
s  f  f  (b  f )  (b  s)  T
Original statement
Definition of implication
Associativity of AND
Re-arranging
s  f  (b  f )  (b  s)  T
Idempotent law
f  (b  f )  s  (s  b)  T
Re-arranging
f  (b  f )  s  T
 f  (b  f )  s  T
( f  b)  ( f  f )  s  T
( f  b)  F   s  T
( f  b)  s  T
f  b  s  T
Absorption law
Re-arranging
Distributive law
Negation law
Domination law
Associativity of AND
48
What if it weren’t possible to assign
such values to s, b, and f?
(s  f )  (b  f )  ( f  (b  s))  s  T
(s  f )  (b  f )  ( f  (b  s))  s  T
Original statement
Definition of implication
... (same as previous slide)
( f  b )  s  s  T
f  b  s  s  T
f b F T
Domination law
Re-arranging
Negation law
f F T
Domination law
F T
Domination law
Contradiction!
49
Functional completeness
• All the “extended” operators have
equivalences using only the 3 basic
operators (and, or, not)
– The extended operators: nand, nor, xor,
conditional, bi-conditional
• Given a limited set of operators, can you
write an equivalence of the 3 basic
operators?
– If so, then that group of operators is
functionally complete
50
Functional completeness of NAND
• Show that | (NAND) is functionally complete
• Equivalence of NOT:
– p | p ≡ p
– (p  p) ≡ p
– (p) ≡ p
Equivalence of NAND
Idempotent law
51
Functional completeness of NAND
• Equivalence of AND:
– p  q ≡ (p | q)
– p|p
– (p | q) | (p | q)
Definition of nand
How to do a not using nands
Negation of (p | q)
• Equivalence of OR:
– p  q ≡ (p  q) DeMorgan’s equivalence of OR
– As we can do AND and OR with NANDs, we can thus
do ORs with NANDs
• Thus, NAND is functionally complete
52