1. Boolean Algebra

2. Outline Laws and theorems of Boolean Algebra Switching functions
Logic functions: NOT, AND, OR, NAND, XOR, XNOR
Switching function representations
Canonical forms
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Boolean Algebra

3. Axiomatic of Boolean Algebra
A Boolean algebra consists of a set B with two binary operations (  “AND”,  “OR”) and a unary operation ( ¯ or  “NOT”), such that the following axioms satisfy:
Set B contains at least two distinct elements a and b.
Closure: For every a, b  B,
a + b  B
a  b  B
Commutative Laws: For every a, b  B,
a + b = b + a
a  b = b  a
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Boolean Algebra

4. Axiomatic of Boolean Algebra
Associative Laws: For every a, b, c  B,
(a + b) + c = a + (b + c)
(a  b)  c = a  (b  c)
Identities: For every a  B,
 an identity element 0, such that a + 0 = a
 an identity element 1, such that a  1 = a
Distributive Laws: For every a, b, c  B,
a  (b + c) = (a  b) + (a  c)
Complement: For each a  B,  an such that
a = 1
a  = 0
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Boolean Algebra

5. Boolean function A Boolean function uniquely maps Bn to B.
A Boolean function uniquely maps Bn to B.
A Boolean expression is an algebraic statement containing Boolean (binary) variables and operators (, +, and ), that is (AND, OR, and NOT)
A literal is a variable itself or its complement. When a Boolean function is implemented with logic gates, each literal represents an input to a gate, and each term is implemented a gate.
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Boolean Algebra

6. Examples F = XYZ F = X + Y Z F = X Y Z + X YZ + XZ
Z = A  B  (C + D)
Z = (A  (B  (C + D)))
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Boolean Algebra

7. Laws and Theorems of Boolean Algebra
Duality Every Boolean expression is deducible from the postulates of Boolean algebra remains valid if the operators and the identity elements are interchanged. That is interchange OR and AND operators and replace 1's by 0's and 0's by 1's.
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Boolean Algebra

8. Examples X + 1 = 1  X  0 = 0 X + XY = X  X(X + Y)
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Boolean Algebra

9. Laws and Theorems Boundness law: A + 1 = 1 A  0 = 0 Identity law:
Idempotent Theorem:
A + A = A
A  A = A
Involution Theorem:
(A) = A
Theorem of complementarity:
A + A = 1
A  A = 0
Commutative law:
A + B = B + A
AB = BA
Associative law:
A + (B + C) = (A + B) + C
A(BC) = (AB)C
Distributive law:
A (B + C) = AB + AC
A + BC = (A+B)(A+C)
DeMorgan's Theorem:
(A + B) = A B
(AB) = A + B
Absorption law:
A + AB = A
A(A + B) = A
Consensus Theorem:
AB+BC+AC = AB+AC
(A+B)(B+C)(A+C) = (A+B)(A+C)
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Boolean Algebra

10. Examples
Simplify the following Boolean expressions to a minimum number of literals
X + X Y  X + Y
XY + X Y  Y
X (X + Y )  XY
X Y Z + X YZ + XY  X Z + XY
XY + X Z + YZ  XY + X Z
(X+Y)(X+Z)(Y+Z)  X Y + XZ
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Boolean Algebra

11. Switching Functions
A switching algebra is a Boolean algebra whose set B contains only two values 0 and 1.
A switching function uniquely maps Bn to B.
f (X, Y, Z) = XY + X Z + YZ
If X = 0, Y = 1, Z = 0, then f (X, Y, Z) = 0.
If X = 0, Y = 1, Z = 1, then f (X, Y, Z) = 1.
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Boolean Algebra

12. Truth Tables
A switching function can be represented as a Boolean function or in a tabular form called truth table. A truth table is a list of possible combinations of inputs that correspond to the values of the switching function (output).
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Boolean Algebra

13. Example Truth table of f (X, Y, Z) = XY + X Z + YZ X Y Z f 11
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Boolean Algebra

14. Switching Functions
There are 16 possible switching functions of two variables: X Y 
XY
X Y  OR
NOR = Y  X 
NAND 1
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Boolean Algebra

15. Canonical and Standard Forms
Minterms
Maxterms
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Boolean Algebra

16. Minterms
For two binary variables A and B combined with an AND operation, the minterms or standard products are: AB, AB, AB, and AB. That is, two binary variables provide 22 = 4 possible combinations (minterms.) n variables have 2n minterms. Each minterm has each variable being primed if the corresponding bit of the binary number is a 0 and unprimed if a 1.
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Boolean Algebra

17. Maxterms
Similarly, two binary variables A and B combined with an OR operation, the maxterms or standard sums are: A+B, A+B, A+B, and A+B. That is, two binary variables provide 22 = 4 possible combinations (maxterms.) n variables have 2n maxterms. Each maxterm has each variable being primed if the corresponding bit of the binary number is a 1 and unprimed if a 0.
A maxterm is the complement of its corresponding minterm, and vice versa.
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Boolean Algebra

18. Boolean function Sum of Products (or Minterms)
Sum of Products (or Minterms)
A Boolean function can be expressed as a sum of minterms. The minterms whose sum defines the Boolean function are those that give the 1's of the function in a truth table.
Product of Sums (or Maxterms)
A Boolean function can be expressed as a product of maxterms. The maxterms whose sum defines the Boolean function are those that give the 0's of the function in a truth table.
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Boolean Algebra

19. Minterms and Maxterms for Three Binary Variables
Input
Minterm
Maxterm X Y Z
Term
Designation
XYZ m0
X+Y+Z M0 1
XYZ m1
X+Y+Z M1
XYZ m2
X+Y+Z M2
XYZ m3
X+Y+Z M3
XYZ m4
X+Y+Z M4
XYZ m5
X+Y+Z M5
XYZ m6
X+Y+Z M6
XYZ m7
X+Y+Z M7
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Boolean Algebra

20. Examples X Y Z Function F1 Function F2 1 CS 3402--Digital Logic1
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Boolean Algebra

21. Examples F1 and F2 can be expressed as a sum of products as follows:
F1 = XYZ+XYZ+XYZ = m1+ m4+ m7
F2 = XYZ+XYZ+XYZ +XYZ = m3+ m5+ m6 + m7
F1 and F2 can also be expressed as a product of sums as follows:
F1 = (X+Y+Z)(X+Y+Z)(X+Y+Z)(X+Y+Z)(X+Y+Z)
= M0 M2 M3 M5 M6
F2 = (X+Y+Z)(X+Y+Z)(X+Y+Z)(X+Y+Z)
= M0 M1 M2 M4
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Boolean Algebra

22. Notation
Boolean functions expressed as a sum of products or product of sums are said to be in canonical form
A convenient way to express these function is by using a short notation, decimal form:
F1(X, Y, Z) = m(1,4,7)
F2(X, Y, Z) = m(3,5,6,7) or
F1(X, Y, Z) =  M(0,2,3,5,6)
F2(X, Y, Z) =  M(0,1,2,4)
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Boolean Algebra

23. Standard forms
A Boolean function is said to be in standard form if the function contains one, two or any number of literals. For example:
F1 = Y+XY+XYZ or
F2 = X(Y+Z)(X+Y+Z+W)
A Boolean function may be expressed in a nonstandard form. For example, the function
F = (WX+YZ)(WX+YZ)
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Boolean Algebra

24. Example 1
1. Given the following truth table. Express F in a canonical minterms and maxterms. X Y Z F 1
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Boolean Algebra

25. Example 2
2. Design a digital logic circuit that will activate an alarm if a door or window is open during non-business hours. Assume that
Clock
C = 0 (non-business hours)
1 (business hours)
Door
D = 0 (closed)
1 (opened)
Window
W = 0 (closed)
Alarm
A = 0 (off)
1 (on)
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Boolean Algebra

26. Conversion between canonical form
To convert from a sum of products to a product of sums: rewrite the minterm canonical form in a shorthand notation then replace the existing term numbers by the missing numbers. For example:
F1(X, Y, Z) = m(1,3,6,7) =  M(0,2,4,5)
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Boolean Algebra

27. Conversion between canonical form
To convert from a product of sums to a sum of products: rewrite the maxterm canonical form in a shorthand notation then replace the existing term numbers by the missing numbers. For example:
F1(X, Y, Z) =  M(0,2,4,5) = m(1,3,6,7)
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Boolean Algebra

28. Conversion between canonical form
To obtain the minterm (or maxterm) canonical form of the complement, given the Boolean function in a sum of products (or product of sums) form : list the term numbers that are missing in For example:
F(X, Y, Z) = m(0,2,4,5)  F(X, Y, Z) = m (1,3,6,7)
F(X, Y, Z) = M(1,3,6,7)  F(X, Y, Z) = M(0,2,4,5)
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Boolean Algebra

29. Don't Care Conditions F(A,B,C,D) = m(1,3,7,11,13,15) + d(0,2,5)
BCD increment by 1 function.
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Boolean Algebra

30. Logic Functions AND Operation Z = X  Y Inputs Output X Y Z 11
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Boolean Algebra

31. Logic Functions OR Operation Z = X + Y Inputs Output X Y Z 11
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Boolean Algebra

32. Logic Functions NOT Operation Z = X  Inputs Output X Z 11
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Boolean Algebra

33. Logic Functions NAND Operation Z = (X  Y) Inputs Output X Y Z 11
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Boolean Algebra

34. Logic Functions NOR Operation Z = (X + Y) Inputs Output X Y Z 11
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Boolean Algebra

35. Logic Functions XOR Operation Z = X  Y Inputs Output X Y Z 11
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Boolean Algebra

36. Logic Functions XNOR Operation Z = (X  Y) Inputs Output X Y Z 11
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Boolean Algebra

37. Switching function representations
There are 3 ways to represent a switching function:
Boolean expression
Truth table
Logic diagram
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Boolean Algebra

38. Positive and Negative Logic
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Boolean Algebra

39. Positive and Negative Logic
Truth Table
Positive Logic
Negative Logic x y z
low 1
high
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Boolean Algebra

40. Positive and Negative Logic
Truth Table
Positive Logic
Negative Logic x y z
low 1
high
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Boolean Algebra

41. Example Example: Traffic lights -- to define three signals
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Boolean Algebra