Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

SECTION B] THE ARITHMETICAL PRODUCT OF A CLASS OF CLASSES

125

Among the more important propositions in this number are the following:
*114'21. F. IlNc''a = Nc'a
I.e. a product of one factor is equal to that factor.
114-23. F: A e K. ). HINc'K = 0
I.e. a product vanishes if one of its factors is zero. The converse requires
the multiplicative axiom, as appears from the proposition
*114'26. F:. Mult ax. _: INc'K = O.K. A K
I.e. the multiplicative axiom is equivalent to the assumption that a
product vanishes when, and only when, one of its factors is zero.
*114'301. F: Kc n X = A. D. Ce'(Kc X) sm efaK X e4X
whence
*114-31. F: K n X = A... HINc'K xc, Nc'X = IINc'(K u X)
which is a form of the associative law, an(l
*114'35. F: a +$ 3.. IDNc'(t'a u t'/) = Nc'a xc Nc'6/
which connects the two sorts of mulltiplication.
*114'41. F: X C 1. D. IlNc'(K u X) = LINcK
I.e. unit factors make no difference to the value of a product.
*114 51. p: T prs X e si sl X. D. (;i Te) r X'X e (FA' ) smn (ef o)
This proposition gives a correlator of EA'K amnd e4'X as a function of
a double correlator of K and X, and thus leads to
*114'52. F: Fc sm sm X. X. IlNcc = HNc'X. ea'c sm ea'
Hence, by the propositions of *111, we infer
*114-571. k:. Mult ax. D: I, v e NC. c, X e,U n Cl1'v... Nc'K = INc'X
I.e. assuming the multiplicative axiom, if K and X each consist of /u
classes of v terms each, their products are equal.
We have next various forms of the associative law, beginning with
*114'6. F: K e Cls2 excl. ) nNc'EA K = HNc's'K
which is an immediate consequence of *85'44. The other form is
*114-632.: S7yel-l1.7C a S. 7yn 7= A..
EiI {(a) -*4 a y *. = a x S'a} sm eC(y u S",y)
As to the sense in which this is a form of the associative law, see the
observations following *114'6.
*114-01. lNc'K = Nc'eS'K Df
*114-1.. H Nc'K = Nc'e~a'K [(*114-01)]
*114-11. F: 3 e nNc'K.. -. Sm Ea'KC. / e NC 'C'K [*114'1. *100-31]
*114'12. F. eC'K e TINC'c [*100'3. *114-1]
*114-2. F. H Nc'A = 1 [*83-15. *101-2]