math

1. BOOLEAN ALGEBRA

2. COURSE CONTENT
1. Basic Operations
2. Boolean expressions and truth table
3. Basic theorems
4. Commutative, Associative, Distributive and
5. DeMorgan’s laws
6. Simplification theorems
7. Multiplying out and factoring expressions
8. Exclusive-OR and Equivalence Operations
9. The Consensus theorems
10.Algebraic simplifications of switching
expressions
11.Proving Validity of an equation

3. 1. Basic Operations
The basic operations of Boolean algebra are conjunction, disjunction,
and negation.
These Boolean operations are expressed with the corresponding binary
operators AND, and OR and the unary operator NOT, collectively
referred to as Boolean operators.

4. 2. Boolean expressions and truth table
Boolean expression are formed by application of the basic operation to
one or more variables or constants.
Simple expression consist of a single constant or variable more
complicated expression are formed by combining two or more other
expression using AND or OR gates.
Expression: AB’+C

5. 2. Boolean expressions and truth table
Expression: AB’+C
A B C B’ AB’ AB’+C
0 0 0 1 0 0
0 0 1 1 0 1
0 1 0 0 0 0
0 1 1 0 0 1
1 0 0 1 1 1
1 0 1 1 1 1
1 1 0 0 0 0
1 1 1 0 0 1

6. 2. Boolean expressions and truth table
Questions:
1. A+B=B+A
2. A’B’+A’+B’
3. D.(A’B+BC)

7. 3. Basic Theorems:
1. Operation with 0 & 1:
 X+0=X
 X+1=X
 X.0=0
 X.1=X
2. Idempotent Law:
 X+X=X
 X.X=X

8. 3. Basic Theorems:
3. Involution Law:
 (X’)’ =X
4. Laws of Complementarity:
 X + X’ = 1
 X . X’ = 0

9. 4. Boolean Laws:
Commutative Law
. Commutative Law of Addition:
A + B = B + A
A B A+B
0 0 0
0 1 1
1 0 1
1 1 1
A B B+A
0 0 0
0 1 1
1 0 1
1 1 1

10. 4. Boolean Laws:
Commutative Law
. Commutative Law of Multiplication:
A . B = B . A
A B A.B
0 0 0
0 1 0
1 0 0
1 1 1
A B B.A
0 0 0
0 1 0
1 0 0
1 1 1

11. 4. Boolean Laws:
Associative Law
. Associative Law of Addition:
A+(B+C) = (A+B)+C
A B C B+C A+(B+C)
0 0 0 0 0
0 0 1 1 1
0 1 0 1 1
0 1 1 1 1
1 0 0 0 1
1 0 1 1 1
1 1 0 1 1
1 1 1 1 1
A B C A+B (A+B)+C
0 0 0 0 0
0 0 1 0 1
0 1 0 1 1
0 1 1 1 1
1 0 0 1 1
1 0 1 1 1
1 1 0 1 1
1 1 1 1 1

12. 4. Boolean Laws:
Associative Law
. Associative Law of Multiplication:
A(BC) = (AB)C
A B C BC A(BC)
0 0 0 0 0
0 0 1 0 0
0 1 0 0 0
0 1 1 1 0
1 0 0 0 0
1 0 1 0 0
1 1 0 0 0
1 1 1 1 1
A B C AB (AB)C
0 0 0 0 0
0 0 1 0 0
0 1 0 0 0
0 1 1 0 0
1 0 0 0 0
1 0 1 0 0
1 1 0 1 0
1 1 1 1 1

13. 4. Boolean Laws:
Associative Law
. Distributive Law of Addition:
A+ (BC) = (A+B)(A+C)
. Distributive Law of Multiplication:
A (B+C) = AB+AC

14. 5. DeMorgan’s laws:
. (A + B) = A B
The Complement of a sum is equal to the product of Complement.
. (AB) = A + B
The Complement of a product is the equal to sum of Complement.
“Make Truth Table”

15. 5. DeMorgan’s laws:
. (A + B) = A B
The Complement of a sum is equal to the product of Complement.
. (AB) = A + B
The Complement of a product is the equal to sum of Complement.
“Make Truth Table”

16. 6. Simplification theorems:
1. Uniting:
(X+Y) (X+Y’) = X
X Y Y’ X+Y X+Y’ (X+Y) (X+Y’)
0 0 1 0 1 0
0 1 0 1 0 0
1 0 1 1 1 1
1 1 0 1 1 1

17. 6. Simplification theorems:
2. Absorption:
X+XY=X
X Y XY X+XY
0 0 0 0
0 1 0 0
1 0 0 1
1 1 1 1

18. 6. Simplification theorems:
2. Absorption:
X(X+Y) = X
X Y X+Y X(X+Y)
0 0 0 0
0 1 1 0
1 0 1 1
1 1 1 1

19. 6. Simplification theorems:
3. Elimination:
. X+X’Y = X+Y
. X(X’+Y) = XY
4. Consensus:
. XY + X’Z + YZ = XY + X’Z
. (X+Y)(X’+Z)(Y+Z) = (X+Y)(X’+Z)

20. 7. Exclusive OR and equivalent operation:
1. XOR:
. Q = (A ⊕ B) = A’.B + A.B’
2. XNOR:
. A ⊙ B = AB + ĀB ̅.

21. 8. Algebraic simplifications of switching expressions:
Expression no 1:
A.B’ + A.B + B.C
A(B’+B) + B.C
A(0+1) + B.C
A.1 + B.C
A+B.C
Expression no 2:
A+A.B
A(1+B)
A.1
A

22. 8. Algebraic simplifications of switching expressions:
Expression no 3:
AB+ A(B+C)+B(B+C)
AB+AB+AC+BB+BC (Distributive Law)
AB+AB+AC+B+BC (BB=B)
AB+AC+B+BC (AB+AB=AB)
AB+AC+B (B+BC=B)
B+AC (AB+B=B)

23. 9. Proving Validity of an equation
. Construct a truth Table and evaluate both side eq.
. Manipulate one side of eq by applying various rules and theorems until its it is identical with other
side.
Expression #01
A.B’+A.B+B.C= A+B.C
Truth Table:
Expression #02
A+A.B=A
Expression #03
A.B’ +A.B + B.C = A+B.C