boolean algebra and logic simplification. boolean addition & multiplication boolean addition...
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Boolean Addition & Multiplication
• Boolean Addition performed by OR gate• Sum Term describes Boolean Addition
• Boolean Multiplication performed by AND gate• Product Term describes Boolean Multiplication
CBA
Boolean Addition
• Sum of literals
• Sum term = 1 if any literal = 1• Sum term = 0 if all literals = 0
BA BA CBA
Boolean Multiplication
• Product of literals
• Product term = 1 if all literals = 1• Product term = 0 if any one literal = 0
BA. BA. CBA ..
Laws, Rules & Theorems of Boolean Algebra
• Commutative Law for addition and multiplication
• Associative Lawfor addition and multiplication
• Distributive Law• Rules of Boolean Algebra• Demorgan’s Theorems
Commutative Law
• Commutative Law for AdditionA + B = B + A
• Commutative Law for MultiplicationA.B = B.A
A
B
A + B A + BB
A
A
B
A.B A.BB
A
Associative Law
• Associative Law for Addition A + (B + C) = (A + B) + C
A
B
A+(B+C)
C
B+C
A
B
C
A+B
(A+B)+C
Associative Law
• Associative Law for MultiplicationA.(B.C) = (A.B).C
A
B
A.(B.C)
C
B.C
A
B
C
A.B
(A.B).C
Rules of Boolean Algebra
1. A + 0 = A2. A + 1 = 13. A.0 = 04. A.1 = A5. A + A = A6. A + = 1
7. A.A = A8. A. = 09. = A10. A + A.B = A11. A + = A + B12. (A+B).(A+C)
= A+B.C
A
A
A
BA.
Demorgan’s Theorems
• Any number of variables
• Combination of variables
ZYXZYX ..
ZYXZYX ..
).().().).(.( BCACBABCACBA
BCACBA .).().(. BCACBA ).().(
CBBACABA ....
CBCABA ...
Simplification using Boolean Algebra
• AB + A(B+C) + B(B+C)= AB + AB + AC + BB +BC= AB + AC + B + BC= AB + AC + B= B + AC
Standard forms of Boolean Expressions
Standardization make the implementation of Boolean expression much more easier and systematic
Two forms are• Sum-of-Products form• Product-of-Sums form
Standard forms of Boolean Expressions
• Sum-of-Products formAB + ABCABC + CDE +
• Product-of-Sums form
DCBACCBABA
))(( CBABA
))()(( DCBEDCCBA
))()(( CACBABA