Boolean Algebra
Logical Statements

A proposition that may or may not be true:



Today is Monday
Today is Sunday
It is raining
Compound Statements

More complicated expressions can be built from
simpler ones:



Today is Monday AND it is raining.
Today is Sunday OR it is NOT raining
Today is Monday OR today is NOT Monday


Today is Monday AND today is NOT Monday


(This is a tautology)
(This is a contradiction)
The expression as a whole is either true or false.
Things can get a little tricky…

Are these two statements equivalent?


It is not nighttime and it is Monday OR it is raining
and it is Monday.
It is not nighttime or it is raining and Monday AND it
is Monday.
Boolean Algebra


Boolean Algebra allows us to formalize this sort
of reasoning.
Boolean variables may take one of only two
possible values: TRUE or FALSE.



(or, equivalently, 1 or 0)
Algebraic operators: + - * /
Logical operators - AND, OR, NOT, XOR
Logical Operators




A AND B is True when both A and B are true.
A OR B is always True unless both A and B are
false.
NOT A changes the value from True to False or
False to True.
XOR = either a or b but not both
Writing AND, OR, NOT





A AND B = A ^ B = AB
A OR B = A v B = A+B
NOT A = ~A = A’
TRUE = T = 1
FALSE = F = 0
Exercise




AB + AB’
A AND B OR A AND NOT B
(A + B)’(B)
NOT (A OR B) AND B
Boolean Algebra



The = in Boolean Algebra means equivalent
Two statements are equivalent if they have the
same truth table. (More in a second)
For example,


True = True,
A = A,
Truth Tables

Provide an exhaustive approach to describing
when some statement is true (or false)
Truth Table
M
R
T
T
T
F
F
T
F
F
M’
R’
MR
M+R
Truth Table
M
R
M’
R’
T
T
F
F
T
F
F
T
F
T
T
F
F
F
T
T
MR
M+R
Truth Table
M
R
M’
R’
MR
T
T
F
F
T
T
F
F
T
F
F
T
T
F
F
F
F
T
T
F
M+R
Truth Table
M
R
M’
R’
MR
M +R
T
T
F
F
T
T
T
F
F
T
F
T
F
T
T
F
F
T
F
F
T
T
F
F
Example


Write the truth table for A(A’ + B) + AB’
Fill in the following columns:

A, B, A’, B’, A’ + B, AB’, A (A’ + B), whole expression.
A (A’ + B) + AB’
A B
A’
B’
A’ + B
A B’
A(A’+B)
Whole
T
T
F
F
T
F
T
T
T
F
F
T
F
T
F
T
F
T
T
F
T
F
F
F
F
F
T
T
T
F
F
F
Exercise

Write the truth table for (A + A’) B
Solution to (A + A’) B
A
B
A’
A + A’
(A + A’) B
T
T
F
T
T
T
F
F
T
F
F
F
T
F
T
T
T
T
T
F
Boolean Algebra - Identities




A + True = True
A + False = A
A+A=A

A+B=B+A
(commutative)



A AND True = A
A AND False = False
A AND A = A
AB = BA
(commutative)
Associative and Distributive
Identities





A(BC) = (AB)C
A + (B + C) = (A + B) + C
A (B + C) = (AB)+(AC)
A + (BC) = (A + B) (A + C)
Exercise: using truth tables prove 
A(A + B) = A
Solution: A AND (A OR B) = A
A
B
A+ B
A (A + B)
T
T
T
T
T
F
T
T
F
T
T
F
F
F
F
F
Using Identities





A + (BC) = (A + B)(A + C)
A(B + C) = (AB) +(AC)
A(A + B) = A
A+A=A
Exercise - using identities prove:
A + (AB) = A
 A +(AB) = (A +A)(A + B)
 = A (A + B) = A

Identities with NOT



(A’)’ = A
A + A’ = True
AA’ = False
DeMorgan’s Laws



(A + B)’ = A’B’
(AB)’ = A’ + B’
Exercise - Simplify the following with identities

(A’B)’
Solving a Truth Table
A B X When you see a True value in the X column,
T T T you must have a term in the expression. Each
term consists of the variables AB. A will be
T F T
NOT A when the truth value of A is False, B
F T F will be NOT B when the truth value of B is
F F F false. They will be connected by OR.
For example,
X = AB + AB’
Exercise: Solving a Truth Table
A B X When you see a True value in the X column,
T T T you must have a term in the expression. Each
term consists of the variables AB. A will be
T F F
NOT A when the truth value of A is False, B
F T T will be NOT B when the truth value of B is
F F F false. They will be connected by OR.
Solve the Truth Table given above.
Exercise: Solving a Truth Table
A B X When you see a True value in the X column,
T T T you must have a term in the expression. Each
term consists of the variables AB. A will be
T F F
NOT A when the truth value of A is False, B
F T T will be NOT B when the truth value of B is
F F F false. They will be connected by OR.
Solution is,
X = AB + A’B = (A AND B) OR ( NOT A AND B)