CHAPTER 2: Boolean Switching Algebra( 布尔开关代数 )

1. CHAPTER 2: Boolean Switching Algebra(布尔开关代数) • Introduction： Boolean algebra is a mathematical system that defines a series of logical operations (AND, OR, NOT) performed on sets of variables (a, b, c,.….. ).When stated in this form , the expression is called a Boolean equation or switching equation.

2. Three Primitive Logic Function--------AND • Operator Symbols: “()”, “ * ”, “ ·”,no space S=x y s=x*y s=(x) (y) • Two input and truth table • Distinctive shape symbols for AND gate S

3. Three Primitive Logic Function--------OR • Operator Symbols: “+” s=x + y • Two input and truth table • Distinctive shape symbols for OR gate S

4. Three Primitive Logic Function--------NOT • Operator Symbols: “ ”，“ ” x=x ，x=x • truth table • Distinctive shape of the NOT symbols ׳ ׳ ׳ x

5. Derived logic functions ------NAND 、NOR、EX-OR、EX-NOR • NAND (not and)：s=(x y)׳ ; s=x y

6. NOR (not or)：s=(x+ y)׳ ;s=x + y Derived logic functions ------NAND 、NOR、EX-OR、EX-NOR

7. EX-OR (exclusive or)：s= x⊕ y Derived logic functions ------NAND 、NOR、EX-OR、EX-NOR S=XY+XY

8. EX-NOR (exclusive not or)：s= x⊕ y ; s=x☉ y Derived logic functions ------NAND 、NOR、EX-OR、EX-NOR S=XY+XY

9. IEEE Logic Symbols • IEEE general logic symbols: Name or Symbol Indicating Function Input Variables Output Variables

10. Z Z Z X X X Y Y Y IEEE Logic Symbols & + >1 Z=X+Y Z=XY X׳ X׳ X X

11. Z Z Z Z Z X X X X X Y Y Y Y Y IEEE Logic Symbols & + Z=X+Y Z=XY ⊕ =1 =1 Z=X☉ Y Z=X ⊕Y

12. The Basic Definition of A Binary Switching Algebra • Set • A set of elements is a collection of items that have something in common. e.g. D={0,1,2,3,4,5,6,7,8,9,} 3∈D denote that element 3 is a member of D P∈D means that P is not a member of set D

13. The Basic Definition of A Binary Switching Algebra • Equivalence • If two variables x and y , have the same value they are said to be equivalent. e.g. X=0 Y=0 X=1 Y=1 } X=Y } X=Y

14. The Basic Definition of A Binary Switching Algebra • closure • A set is closed with respect to a binary operator(· , +)if, when the operation is applied to members of the set, the result is a member of the set. The property of closure would not hold if an operation could produce a result not in the initial set • Let B={0,1} Truth table illustrating closure for OR Truth table illustrating closure for AND

15. The Basic Definition of A Binary Switching Algebra • Identity • A binary operator(· , +)has an identity element we will call Ie. Ie must be contained in the binary number set {0,1}. When Ie is ANDed with a variable x the result is x; When Ie is ORed with a variable x the result is x. x Ie =x x +Ie =x OR identity AND identity

16. 2.2 Switching Algebra Properties 1、Commutative(交换) Properties : X+Y=Y+X X·Y=Y·X 2、Associative(结合) Properties: (X+Y)+Z=X+(Y+Z) (X·Y) ·Z=X·(Y·Z) 3、Distributive(分配) Properties: X+(Y·Z)=(X+Y) ·(X+Z) X·(Y+Z)=X·Y+X·Z 4、Identity (0-1律) Properties: X+0=X X·1=X X+1=1 X·0=0 5、Complement (互补) Properties: X+X=1 X·X=0

17. 2.2 Switching Algebra---Theorems • Binary Variables and Constants： 0+0=0 1+0=1 0+1=1 1+1=1 0·0=0 1·0=0 0·1=0 1·1=1 2、Idempotency Property (幂等性): X+X=X X·X=X 3、Absorption Property(吸收): X+X·Y=X X·(X+Y)=X X+X ·Y=X+Y X·(X+Y)=X·Y 4、DeMorgan Properties(反演，狄*摩根): X+Y=X ·Y X·Y=X+Y 5、Adjacency Properties: X·Y+X·Y=X (X+Y) ·(X+Y)=X

18. 2.2 Switching Algebra---Theorems • Idempotence Property: X+X=X X·X=X Proof: X+X =(X+X) ·1 =(X+X) ·(X+X’) =X+(X·X’) =X+0 =X

19. 2.2 Switching Algebra---Theorems Absorption Property: X+X·Y=X X·(X+Y)=X X+X ·Y=X+Y X·(X+Y)=X·Y Proof: X+X·Y=X·1+X·Y =X·(1+Y) =X(Y+1) =X·1 =X X+X ·Y=(X+X) ·(X+Y) (分配) =1·(X+Y) =X+Y

20. 2.2 Switching Algebra---Theorems DeMorgan Properties:X+Y=X ·Y X·Y=X+Y Proof: ∵ (X ·Y)+(X+Y)=(X·Y+X)+Y(结合) =(Y+X)+Y(吸收) =X+(Y+Y) (结合) =X+1 =1 ∵ (X ·Y) ·(X+Y)=X ·Y ·X+X·Y ·Y =0+0 =0 ∴ X+Y=X ·Y (Complement Properties)

21. 2.2 Switching Algebra---Theorems • DeMorgan Properties: x1x2x3…xn=x1+x2+x3+…+xn x1+x2+x3+…+xn=x1.x2.x3…xn

22. 2.2 Switching Algebra---Theorems Adjacency Properties:X·Y+X·Y=X (X+Y) ·(X+Y)=X Proof: X·Y+X·Y=X·(Y+Y) =X·1 =X

23. e.g.1 Use DeMogan’s Theorem to find an equivalent to G=xy+xz. Solution 1. Start with the complete function: a. Let A=xy and B=xz b. Then G=A+B c. Complementing step b, G=(A’B’)’ 2.Having converted the OR to AND, we can now proceed to convert terms xy and xz a. Converting, xy=(x’+y’)’ b. converting, xz=(x’+z’)’ c. Therefore, G=[(x’+y’)(x’+z’)]’

24. e.g.2 Find the DeMogan equivalent to equation F=x (y+ z’)’. Solution 1. By treating (y+z’)’ as a unit the conversion yields F=[x’+ (y+ z’)’’)]’=(x’+y+z’)’ 2. By applying DeMorgan’s theorm once more, a single three-input AND gate equivalent can also be found F=(x’+y+z’)’=xy’z

25. 2.3 Functionally Complete Operation Sets(完备运算集) • Functionally Complete Operation Sets is a set of logic functions from which any combinational(组合) logic expression can be realized. • FC1={AND,OR,NOT} • FC2={NOR} • FC3={NAND} • FC4={EXOR,AND}

26. xy=xy, x+y = xy, x=xx • xy=x+y, x+y = x+y, x=x+0 • x=1⊕x, x+y=x⊕y⊕xy

27. 2.4 Reduction Of Switching Equations Using Boolean Algebra • Why reduction? to reduce cost of the circuit • “And-Or” expression (sum of products) • “Or-And” expression (product of sums) eg: F(A,B,C)=A’+BC’+AB’C F(A,B,C,D)=(A+B)(C+D’)(A’+B+C)

28. 2.4 Reduction Of Switching Equations Using Boolean Algebra • “And-Or” expression (sum of products) E.g. F=AC+ABC+ACD+CD Solution: F=AC(1+B)+CD(1+A) =AC+CD P72

29. 2.4 Reduction Of Switching Equations Using Boolean Algebra • “And-Or” expression Exe. 1、 F=AC(B+BD)+ACD 2、 F=(A⊕B)AB+AB+AB 3、F=(A+B+C)(A+B+C) ANS:1、 ABC+AD 2、A+B 3、AC+AB+BC

30. 2.4 Reduction Of Switching Equations Using Boolean Algebra • “Or-And” expression (product of sums) E.g. S=A(B’+C)’(BC)’ Solution: S=A(BC’)(B’+C’) DEMO =(ABC’)(B’+C’) Distribution = ABB’C’+ABC’C’ .. =ABC’ complement, idempotence P72

31. 2.4 Reduction Of Switching Equations Using Boolean Algebra • “Or-And” expression (product of sums) Exe. 1、 F=(A+B)(A+B)(B+C)(B+C+D) 2、 F=(A+B)(A+B)(B+C)(A+C) ANS:1、A(B+C) 2、(A+B)(A+B)C P72

32. 2.5 Realization of Switching Functions • There are three ways to describe the switching functions • Switching equations • Truth table • Logic diagram • How to solve a combinational logic function? • Problem statement • Construct truth table • Logic expression • Draw the diagram using logic symbols

33. a’ a a’b a’b+ab’ b ab’ b’ 2.5.1 Conversion of Switching Functions to Logic Diagram • E.g. Draw the logic diagram for T=a’b+ab’ • Share common terms when multiple output equations (P75, ex. 2.13 )

34. 2.5.2 Converting Logic Diagram to Switching Equations • To Analyze logic diagrams to determine their purpose when troubleshooting or modifying logic designs. • P78, Ex.2.15

35. Homework • Page 83: 24, 26, 29

36. TO BE CONTINUED