Boolean switching algebra B 布尔开关代数 Chapter 2 Basic concept Binary logic function
Boolean switching algebra 布尔开关代数 Chapter 2 Basic concept & Binary logic function
Basic Cancer R Boolean algebra(Logic algebra) is a closure mathematical system that defines a series of logic operation (and, or, not) performed on set k of variables(a, b, c.) which can only have two values of o or o Notated as L=你k,+,°-0,1} Closure(封闭) A set is closed with respect to a operator if, the operation is applied to members of the set, the result is also a member of the set
Basic Concept Boolean algebra (Logic algebra) is a closure mathematical system that defines a series of logic operation (and, or, not) performed on set k of variables (a, b, c …) which can only have two values of 0 or 1. Notated as L={k, +, •, -, 0, 1} Closure(封闭） A set is closed with respect to a operator if, the operation is applied to members of the set, the result is also a member of the set
Basic Concept Commutative properties(交换律) A+B=B+A: AB=BA Associative properties(结合律) (AB) C=A(B C):(A+B)+C=A+(B+C Distributive properties(分配率) A·(B+C)=AB+AC:A+B·C=(A+B)(A+C Complement properties(互补律) A·A'=0;A+A=1 K Identity properties (0-14) A+0=A;A-1=A;A+1=1:A0=0 idempotency property(等幂律) A+A=A:A·A=A Absorption property(吸收律) A+AB=A: A (A+B)=A
Basic Concept Commutative properties (交换律） A+B=B+A ; AB=BA Associative properties （结合律） (A•B)•C=A•(B•C) ; (A+B)+C=A+(B+C) Distributive properties（分配率） A•(B+C)=A•B+A•C ; A+B•C=(A+B)•(A+C) Complement properties（互补律） A•A’=0 ; A+A’=1 Identity properties （0－1 律） A+0=A ; A•1=A ; A+1=1 ; A•0=0 Idempotency property （等幂律） A+A=A ; A•A=A Absorption property（吸收律） A+A•B=A ; A•(A+B)=A
Basic avert R Duality property o Duals are opposites or mirror images of original operators or constants w Operator and dual or w Operator or dual and 2 Constant 1 dual O 2 Constant 0 dual 1
Basic Concept Duality property Duals are opposites or mirror images of original operators or constants. Operator and dual or Operator or dual and Constant 1 dual 0 Constant 0 dual 1
Basic Cancer R Some more important Boolean identities and theorems for convenient referral 2 A+AB=A+B, A(A'+B)=AB 顶A"=A D(A+B)=A'B':(AB)=A'+B a Demorgan's theorems 圆(A1+A2+…+A1+An)=A1'A2…A….An 0(A1A2…A1…An)=A1+A2+…+A1+.+An R AB+AB=A:(A+B)(A+B)=A o AB+A'C+BC=AB+A'C (A+B)(A+C)(B+C)=(A+B)(A+C)
Basic Concept Some more important Boolean identities and theorems for convenient referral A+A’B=A+B ; A(A’+B)=AB A’’=A; (A+B)’=A’B’ ; (AB)’=A’+B’ Demorgan’s theorems (A1+A2+…+Ai…+An )’= A1 ’•A2 ’•…•Ai ’•…•An ’ ( A1 •A2 •…•Ai •…•An )’ =A1 ’+A2 ’+…+Ai ’+…+An ’ AB+AB’=A ; (A+B)(A+B’)=A AB+A’C+BC=AB+A’C ; (A+B)(A’+C)(B+C)=(A+B)(A’+C)
Basic Cancer R A+AB=A+B R A+AB =A1+A'B (Identity) =A(1+B)+AB (Identity) =A1+AB+A'B (Distributive =A+B(A+A) (Distributive) =A+B1 (identity) =A+B
Basic Concept A+A’B=A+B A+A’B =A1+A’B (Identity) =A(1+B)+A’B (Identity) =A1+AB+A’B (Distributive) =A+B(A’+A) (Distributive) =A+B1 (identity) =A+B
Basic avert R AB+A'C+BC=AB+A'C R AB+A'C+BC =AB+A'C+BC(A+A) = AB+ABC+A'C+A'CB =AB(1+C)+AC(1+B) =AB1+A'C1 =AB+A'C
Basic Concept AB+A’C+BC=AB+A’C ; AB+A’C+BC =AB+A’C+BC(A+A’) =AB+ABC+A’C+A’CB =AB(1+C)+A’C(1+B) =AB1+A’C1 =AB+A’C
Basic Cancer & I Substitute theorems If replacing all variable A in a logic equation with a logic function F, the equation would keep in equivalent F1(a1, a2.am)=F2(a1, a2.am) substitute f(x1.xn) for ai: F1(a1, a2..f,am) F2(a1a2…,f,am Ex。A(B+C)=AB+AC R substitute A+D for A (A+D)(B+C=(A+D)B+(A+D)C
Basic Concept Substitute theorems If replacing all variable A in a logic equation with a logic function F ,the equation would keep in equivalent. F1(a1,a2……am)= F2(a1,a2……am) substitute f(x1…xn) for ai: F1(a1,a2…,f,…am)= F2(a1,a2…,f,…am) Ex . A(B+C)=AB+AC substitute A+D for A: (A+D)(B+C)=(A+D)B+(A+D)C
Basic Cancer 工 nverse theorems The complement of any switching function can be found by replacing every variable with its complement, each And with Or, and each Or with Andi constants are replaced by their complement o is replaced by 1 and 1 by O); the original logic operation order should be retained F=G: F=G EXF≡AB+AC F=AB+AC R Demorgan's theorems Inverse theorems F=(AB+AC)=(AB)(AC) F’=A+BA+C? (A+B)(A+C) F=(A+B)(A'+C)
Basic Concept Inverse theorems The complement of any switching function can be found by replacing every variable with its complement, each And with Or, and each Or with And; constants are replaced by their complement (0 is replaced by 1 and 1 by 0) ;the original logic operation order should be retained. F=G ; F’=G’ Ex. F=AB+AC Demorgan’s theorems F’=(AB+AC)’=(AB)’(AC)’ =(A’+B’)•(A’+C’) Ex. F=AB+AC Inverse theorems: F’=A’+B’•A’+C’ ? F’=(A’+B’)•(A’+C’)
Basic Cancer Dual theorems The dual of any switching function can be found by replacing each And with Or, and each Or with And: constants are replaced by their complement (0 is replaced by 1 and 1 by O), and the logic operation order should hold in original R F=G: Fd=Gd EX F=AB+AC Fd=(A+ B)(A+C)
Basic Concept Dual theorems The dual of any switching function can be found by replacing each And with Or, and each Or with And; constants are replaced by their complement (0 is replaced by 1 and 1 by 0) ;and the logic operation order should hold in original. F=G; Fd=Gd Ex. F=AB+AC Fd=(A+B)•(A+C)