### Chapter 4

```Digital Fundamentals
CHAPTER 4
Boolean Algebra and Logic Simplification
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Slide 1
Boolean Operations and Expressions
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Slide 2
Boolean Operations and Expressions
• Multiplication
0+0=0
0+1=1
1+0=1
1+1=1
0*0=0
0*1=0
1*0=0
1*1 =1
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Slide 3
Laws and Rules of Boolean Algebra
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Slide 4
Laws Boolean Algebra
• Commutative Laws
• Associative Laws
• Distributive Law
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Slide 5
Laws of Boolean Algebra
A+B=B+A
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Slide 6
Laws of Boolean Algebra
• Commutative Law of Multiplication:
A*B=B*A
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Slide 7
Laws of Boolean Algebra
A + (B + C) = (A + B) + C
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Slide 8
Laws of Boolean Algebra
• Associative Law of Multiplication:
A * (B * C) = (A * B) * C
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Slide 9
Laws of Boolean Algebra
• Distributive Law:
A(B + C) = AB + AC
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Slide 10
Rules of Boolean Algebra
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Slide 11
Rules of Boolean Algebra
• Rule 1
OR Truth Table
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Slide 12
Rules of Boolean Algebra
• Rule 2
OR Truth Table
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Slide 13
Rules of Boolean Algebra
• Rule 3
AND Truth Table
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Slide 14
Rules of Boolean Algebra
• Rule 4
AND Truth Table
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Slide 15
Rules of Boolean Algebra
• Rule 5
OR Truth Table
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Slide 16
Rules of Boolean Algebra
• Rule 6
OR Truth Table
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Slide 17
Rules of Boolean Algebra
• Rule 7
AND Truth Table
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Slide 18
Rules of Boolean Algebra
• Rule 8
AND Truth Table
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Slide 19
Rules of Boolean Algebra
• Rule 9
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Slide 20
Rules of Boolean Algebra
• Rule 10: A + AB = A
AND Truth Table OR Truth Table
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Slide 21
Rules of Boolean Algebra
• Rule 11:
A  AB  A  B
AND Truth Table OR Truth Table
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Slide 22
Rules of Boolean Algebra
• Rule 12: (A + B)(A + C) = A + BC
AND Truth Table OR Truth Table
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Slide 23
DeMorgan’s Theorem
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Slide 24
DeMorgan’s Theorems
• Theorem 1
XY  X  Y
• Theorem 2
X  Y  XY
Remember:
“Break the bar,
change the sign”
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Slide 25
Standard Forms of Boolean Expressions
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Slide 26
Standard Forms of Boolean Expressions
• The sum-of-product (SOP) form
Example: X = AB + CD + EF
• The product of sum (POS) form
Example: X = (A + B)(C + D)(E + F)
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Slide 27
The Karnaugh Map
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Slide 28
The Karnaugh Map
3-Variable Example
3-Variable Karnaugh Map
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Slide 29
The Karnaugh Map
4-Variable Karnaugh Map
4-Variable Example
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Slide 30
The Karnaugh Map
5-Variable Karnaugh Mapping
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Slide 31
VHDL
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Slide 32
VHDL
• VHDL Operators
• VHDL Elements
and
or
not
nand
nor
xor
xnor
entity
architecture
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Slide 33
VHDL
• Entity Structure
Example:
entity AND_Gate1 is
port(A,B:in bit:X:out bit);
end entity AND_Gate1
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Slide 34
VHDL
• Architecture
Example:
architecture LogicFunction of AND_Gate1 is
begin
X<=A and B;
end architecture LogicFunction
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Slide 35
Hardware Description Languages (HDL)
• Boolean Expressions in VHDL
AND
OR
NOT
NAND
NOR
XOR
XNOR
X <= A and B;
X <= A or B;
X <= A not B;
X <= A nand B;
X <= A nor B;
X <= A xor B;
X <= A xnor B;
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