98 BELL SYSTEM TECHNICAL JOURNAL 



and Sk{Xi , Z2 , ■ • ■ , A^„) is the symmetric function of Xy , -Y2 , • • • , X,^ 

 ivith k for its only a-numhcr. 



This theorem follows from the fact that since / is symmetric in -Yi , A'2 , 

 • • • , A"„, the value of / depends only on the number of JY's that are zero and 

 the values of the I"s. If exactly K of the X's are zero the value of / is 

 therefore /a- , but the right-hand side of (6) reduces to Jk in this case, since 

 then Sj{Xi , A^2 , • • • , X„) = l,j^K, and Sk = 0. 



The expansion (6) is of a form suitable for our design method. We can 

 realize the disjunctive functions Sk{Xi , X2 , • • • , A^,,) with the symmetric 

 function lattice and continue with the general tree network as in Fig. 24, 

 one tree from each level of the symmetric function network. Stopping the 

 trees at F«_i , it is clear that the entire network is disjunctive and a second 

 application of Theorem 1 allows us to complete the function/ with two ele- 

 ments from Yn ■ Thus we have 



Theorem 16. A ny function of m -\- n variables symmetric in m of them can 

 be realized ivith not more than the smaller of 



im + 1)(X(//) + m) or {m + 1)(2" + m - 2) + 2 



elements. In particidar a function of n variables symmetric in n — 2 or more 

 of them can be realized with not more than 



n- - n-\- 2 



elements. 



If the function is symmetric in Xi , X2 , • • • , X„, , and also in Fi , F2 , • • • , 

 Yr , and not in Zi , Z2 , • • • , Z„ it may be realized by the same method, 

 using symmetric function networks in place of trees for the F variables. 

 It should be expanded first about the A''s (assuming m < r) then about the 

 F's and finally the Z's. The Z part will be a set of (w + l)(r + 1) trees. 



References 



1. G. Birklioff and S. MacLane, "A Surve}- of Modern Algebra," Macmillan, 1941. 



2. L. Couturat, "The Algebra of Logic," Open Court, 1914. 



3. J. H. Woodger, "The Axiomatic Method in Biolog}'," Cambridge, 1937. 



4. W. S. McCulloch and VV. Pitts, "A Logical Calculus of the Ideas Immanent in Nervous 



Activity," BuU. Matti. Bioptiysics, V. 5, p. 115, 1943. 



5. E. C. Berkelev, "Boolean Algebra and Applications to Insurance," Record {American 



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6. C. E. Shannon, "A Symbolic Analj'sis of Relay and Switching Circuits," Trans. A. 



1. E. E., V. 57, p. 7"13, 1938. 



7. J. Riordan and C. E. Shannon, "The Number of Two-Terminal Series Parallel Net- 



works," Journal of Matliematics and Pliysics, V. 21, No. 2, p. 83, 1942. 



8. A. Nakashima, Various papers in Nippon Electrical Communication Engineering, A^n\, 



Sept., Nov., Dec, 1938. 



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V. 5," No. 3, p. 98, 1940. 



