BOOLEAN ALGKHHA AND C'lllCUIT DESIGN 



287 



as to tho profei'able combiiiiiijj; forms and oftcMi iiulicajcs additional 

 maiiii)ulations tliat ini^lit l)(> desii'ahlo. 



Hiis network (l(>\(>Iopin('nt is a conihinat ion of mathematics and into- 

 jii'ation l)y inspection. It is cliai'actcrizcd by repeated trials of altei'nati\-e 

 foi'nis and at no sta^e is there any definite assurance* that the optimum 

 ciicuit lias h(HMi attaiiUMJ. Ilo\ve\-er, tiie ease of manipulation proxided 

 t)y the ali>;el)ra f>reatly eiihanc(\s the pi-ohahility of desi<;iiin<i; a hettei- 

 circuit than would be possible by inspection alone. In combining the 

 two-terminal networks, cai'e must be taken not to introduce "sneak" 

 paths which imi)i-()p(>rly connect outputs together. The alg(>bra usually 

 offers means of introducing one or two additional contacts which permit 

 combining networks and y(*t eliminate the adx'erse effects of the sneak 

 paths. 



The abox'c pi-ocedui'e will now be cai'ried out in detail with the switch- 

 ing algebra theorems that ai'e used in all the following algebraic manipu- 

 lations noted at the margin by the number which corresponds to the 

 numl)er of the theorem in the complete listing in Table II. This table is 



Table II 

 Switching (Boolean) Algebra* 



Definitions 



Postulates 



Addition 



(+) = ,LY/) = Series 

 Multiplication 



^.) = OR = Parallel 



Circuit States 



= Closed Circuit 



1 = Open Circuit 



(1) X = or X = 1, where X is a con- 

 tact or a network. 

 (2a) 0-0=0 

 (21)) 1 + 1 = 1 

 (3a) 1-1 = 1 

 (3b) + = 

 (4a) 1-0 = 0-1 = 

 (4b) + 1 = 1+0=1 



Theorems 



(8a) X' + X = 1 



(8b) X'X = 



(9a) + X = X 



(9b) 1-X = X 



(10a) 1 + X = 1 



(10b) 0-X = 



(11a) (X + }'')}' = XY 



(lib) XY' + Y = X + Y 



(12a) (X + F)(X' + Z){Y + Z) 



= (X + Y){X' + Z) 

 (12b) XZ + X'Y + YZ = XZ + X'}' 

 (13) (X + F)(X' + Z) = XZ + X'l' 

 (14a) /(X) = A-/(X)a = i.a' = o 



+ A'-/(X)a=o. a' = i 

 (14b) /(X) = [A +/(X)a.o.a' = .] 



[A' +/(X)a=,.a' = o] 



* Reprinted from The Design of Switching Circuits by Keister, Ritchie and 

 Washburn with the permission of D. Van Nostrand Co., Inc. 



