Computer Science 210Computer Organization

Introduction to Boolean Algebra

George BooleEnglish mathematician (1815-1864)

Boolean algebra Logic Set Theory Digital circuits Programming: Conditions in while and if statements

Boolean ConstantsIn Boolean algebra, there are only two constants, true and false

Boolean constantBinary digitState of a switchVoltage leveltrue1On+5Vfalse0Off0V

Boolean VariablesBoolean variables are variables that store values that are Boolean constants.Let A be true

Let B be false

Etc.

Boolean Operator ANDIf A and B are Boolean variables (or expressions) then A AND B is true if and only if both A and B are true.

Boolean Operator ANDIf A and B are Boolean variables (or expressions) then A AND B is false if and only if either A or B are false or theyre both false.

Boolean Operator ANDWe denote the AND operation like multiplication in ordinary algebra: AB or A.B

Boolean Operator ORIf A and B are Boolean variables (or expressions) then A OR B is true if and only if at least one of A and B is true.

Boolean Operator ORIf A and B are Boolean variables (or expressions) then A OR B is false if and only if both A and B are false.

Boolean Operator ORWe denote the OR operation like addition in ordinary algebra: A+B

Boolean Operator NOTIf A is a Boolean variable (or expression) then NOT A has the opposite value from A.

Boolean Operator NOTWe denote the NOT operation by putting a bar over the variable (or expression)

_ A

Boolean Expressions As with ordinary algebra, a Boolean expression is a well-formed expression made from Boolean constants Boolean variables Operators AND, OR and NOTParenthesesExample: __ ____ AB + (A+C)B

Evaluating a Boolean expressionAt any time, the value of a BE can be computed using the current values of the variables. __AB + (CD)

Let A = trueLet B = falseLet C = trueLet D = false

Then the resulting value is true

Operator precedenceNOT comes first, then AND, and finally OR

(Like arithmetic negation, product, and addition) A + BC is not the same as

(A + B)C

Evaluating a Boolean expressionUnlike ordinary algebra, for a BE, there are only finitely many possible assignments of values to the variables; so, theoretically, we can make a table, called a truth table, that shows the value of the BE for every possible set of values of the variables.

For convenience, use 0 = false 1 = true

Truth Table for AND

ABAB000010100111

Truth Table for OR

ABA+B000011101111

Truth Table for NOT

A_A0110

Filling in a Truth TableIf there are N variables, there are 2N possible combinations of values

Thus, there are 2N rows in the truth table

Fill in the values by counting up from 0 in binary

Construct a truth table for _ ___ E = AB + (A+C)B Example

_ ___ E = AB + (A+C)BAssign the values of the variables first

ABC000001010011100101110111

_ ___ E = AB + (A+C)BThen add columns for each operation

ABC000001010011100101110111

_B11001100

_ ___ E = AB + (A+C)B

ABC000001010011100101110111

_B11001100

_AB00001100

_ ___ E = AB + (A+C)B

ABC000001010011100101110111

_B11001100

_AB00001100

A+C01011111

_ ___ E = AB + (A+C)B

ABC000001010011100101110111

_B11001100

_AB00001100

A+C01011111

___(A+C)10100000

_ ___ E = AB + (A+C)B

ABC000001010011100101110111

_B11001100

_AB00001100

A+C01011111

___(A+C)10100000

___(A+C)B00100000

_ ___ E = AB + (A+C)B

ABC000001010011100101110111

_B11001100

_AB00001100

A+C01011111

___(A+C)10100000

___(A+C)B00100000

E00101100

Laws of Boolean Algebra

Identity

Zero Element

A+0=A, A.1=A

A.0=0, A+1=1

Idempotent

Commutative

A+A=A, AA=A

A+B=B+A, AB=BA

Associative

Distributive

(A+B)+C=A+(B+C)

A(B+C)=AB+AC

(AB)C=A(BC)

A+BC=(A+B)(A+C)

Laws of Boolean Algebra

Absorption

DeMorgan

A+AB=A,

EQ \O(A+B,\S\UP25(____)) = EQ \O(A,\S\UP25(_))

EQ \O(B,\S\UP25(_)) ,A(A+B)=A

EQ \O(AB,\S\UP25(___)) = EQ \O(A,\S\UP25(_)) + EQ \O(B,\S\UP25(_))

Complement

Double Complement

A+ EQ \O(A,\S\UP25(_)) =1, A EQ \O(A,\S\UP25(_)) =0

EQ \O(A,\S\UP25(_),\S\UP30(_)) = A

Boolean Expression Simplification

(A+ EQ \O(B,\s\up30(_)) )C + A EQ \O(C,\s\up30(_)) + ( EQ \O(B+C,\s\up30(____)) )

= (A+ EQ \O(B,\s\up30(_)) )C + A EQ \O(C,\s\up30(_)) + EQ \O(B,\s\up30(_))

EQ \O(C,\s\up30(_)) (DeMorgan)

= (A+ EQ \O(B,\s\up30(_)) )C + (A+ EQ \O(B,\s\up30(_)) ) EQ \O(C,\s\up30(_)) (Distributive)

= (A+ EQ \O(B,\s\up30(_)) )(C+ EQ \O(C,\s\up30(_)) )

(Distributive)

= (A+ EQ \O(B,\s\up30(_)) )1

(Complement)

= A+ EQ \O(B,\s\up30(_))

(Identity)

Boolean Expression Simplification

EQ \O(A,\s\up30(_))

EQ \O(B,\s\up30(_)) + A EQ \O(B,\s\up30(_)) + AB

= EQ \O(A,\s\up30(_))

EQ \O(B,\s\up30(_)) + A EQ \O(B,\s\up30(_)) + A EQ \O(B,\s\up30(_)) + AB (Idempotent)

= EQ \O(B,\s\up30(_)) ( EQ \O(A,\s\up30(_)) +A) + A( EQ \O(B,\s\up30(_)) +B) (Distributive)

= EQ \O(B,\s\up30(_)) 1 + A 1

(Complement)

= EQ \O(B,\s\up30(_)) + A

(Identity)