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F =∑X,Y,Z(0,3,4,6,7) =X’.Y’. Z’ + X’. Y . Z + X . Y’. Z’ + X . Y .Z’ + X .Y . Z
∑X,Y,Z(0,3,4,6,7): is a minterm list . (最小项列表)
The minterm list is also known as the on-set for the logic function. (开集)
F’= [A’ •(B’+C)]’ •[D’+(E’ •F)’]
[F(X1,X2,…,Xn，+，.,’)]’ =FD(X1’,X2’,…,Xn’,+,.,’)
4.1.6 Standard Representations of Logic Functions (P196)
Truth table
真值表
G A, B ,C ( 3,5,6) F ' F A, B ,C ( 3,5,6) F A, B ,C (0,1,2,4,7)
(A’·B·C)’ = A+B’+C’ (A·B’·C)’ = A’+B+C’ 标号互补
Mi = mi’
mi = Mi’
(A·B·C’)’ = A’+B’+C
Example :
F=(A+BC’)’+D•(E+F’)’
F’=?
FD=?
Review of the last class Example:
F=(A+BC’)’+D•(E+F’)’
F’= [A’ •(B’+C)]’ •[D’+(E’ •F)’]
FD= [A •(B+C’)]’ •[D+(E •F’)’]
Standard Representations of Logic Functions
逻辑函数的的标准形式
1、canonical sum (标准和)
The canonical sum of a logic function is a sum of the minterms corresponding to truth-table rows (input combinations) for which the function produces a 1 output.
(最小项编号) (P198)
Minterms (最小项)
A B C
Product Term (乘积项) A’·B’·C’ A’·B’·C A’·B·C’ A’·B·C A·B’·C’
—— An n-variable Minterm is a
normal product term with n literals (n个因子的标准乘积项)
A+B+C’
A+B’+C A+B’+C’ A’+B+C A’+B+C’ A’+B’+C A’+B’+C’
a maxterm can be defined as a sum term that is 0 in exactly one row of the truth table.
1 1 0
1 1 1
class-exercises
FD = ?
1 、F=(A+B+D’) •(A+B’+D)• (A+B+D’) •(A’+C+D) •(A’+C+D’)
2、If F A, B ,C (3,5,6)， G A, B ,C (0,1,2,4,7), what is the relation between the fonction F and fonction G, complement or duality?
+
Standard Representations of Logic Functions
逻辑函数的的标准形式
2、canonical product(标准积)
The canonical product of a logic function is a product of the maxterms corresponding to input combinations for which the function produces a 0 output.
There are 2n such product terms
0 0 0
0 0 1 0 1 0
(n变量函数具有2n个最小项)
0 is the product of Any two
0 1 1
1 0 0 1 0 1 1 1 0 1 1 1
different minterms.
A·B’·C
A·B·C’ A·B·C
标准项
—— 最小项之和 —— 最大项之积
canonical sum 标准和 canonical product 标准积
Minterms
乘积项 (Product Term ) A
A’·B’·C’ A’·B’·C A’·B·C’ A’·B·C A·B’·C’
(最小项) (P197)
A B C
0 0 0
Maxterms (最大项)
—— An n-variable maxterm is a
normal sum term with n literals. A B 0 0 0 0 0 1 0 1
求和项 (n变量最大项是具有n个因子的标准 C (Sum Term) 求和项) 0 A+B+C There are 2n such maxterms. 1 A+B+C’ (n变量函数具有2n个最大项) 0 A+B’+C Any two different sum terms 1 A+B’+C’
F
A’=A B’=B
Negative-Logic (负逻辑)： F = A+B
A B
1 1 0 0 1 0 1 0
F
1 1 1 0
F = F’
G1
F
A B
[F(X1,X2,…,Xn，+，.,’)]’ = FD(X1’,X2’,…,Xn’,+,.,’)
Review of the last class
1. A truth table. 2. An algebraic sum of minterms, the canonical sum. 3. A minterm list using the ∑ notation. 4. An algebraic product of maxterms, the canonical product. 5. A maxterm list using the ∏ notation.
0 0 0
A’·B’·C
A’·B·C’ A’·B·C A·B’·C’ A·B’·C A·B·C’ A·B·C
0 0 1
0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1
1
2 3 4 5 6 7
m4
m5 m6 m7
A’+B+C
A’+B+C’ A’+B’+C A’+B’+C’
M4
M5 M6 M7
product term 乘积项
sum term 求和项
sum-of-products expression “积之和”表达式
product-of-sums expression “和之积”表达式
n-variable minterm n-variable maxterm normal terms n 变量最小项 n 变量最大项
The relationship of Positive-Logic Convention and Negative-Logic Convention are Duality
[F(X1,X2,…,Xn，+，.,’)]’ = FD(X1’,X2’,…,Xn’,+,.,’) Positive-Logic Electrical Function Table (电气功能表) A B L L H H A B L H L H F L L L H (正逻辑)： F = A·B A B 0 0 1 1 0 1 0 1 F 0 0 0 1
(閉集)
Why the canonical product of a logic
is a product of the maxterms?
A 0 真 0 0 值 0 1 表 1 1 1
B 0 0 1 1 0 0 1 1
C 0 1 0 1 0 1 0 1
F 0 1 1 0 = 1 1 1 0
F1
0 1 1 1 1 1 1 1
example
F1 = (A,B,C) ( 1, 5, 7)
What is the duality of F1? F1D = (A,B,C) ( 0, 2, 6 )
(
m i )D = M (2 n -1) - i
Relationship of Minterm and Maxterm
① Mi = mi’ ； mi = Mi’ ； ② 某逻辑函数 F，若用 P 项最小项之和表示， 则其反函数 F’ 可用 P 项最大项之积表示，两
Why the canonical sum of a logic is a
sum of the minterms? A 0 真 0 0 值 0 1 表 1 1 1 B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 F 0 0 0 1 = 0 0 1 1 F1 0 0 0 1 0 0 0 0 F2 0 0 0 0 + 0 0 1 0 F3 0 0 0 0 0 0 0 1
