By G. B Keene

This textual content unites the logical and philosophical points of set idea in a way intelligible either to mathematicians with no education in formal common sense and to logicians and not using a mathematical history. It combines an straightforward point of remedy with the top attainable measure of logical rigor and precision. 1961 variation.

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**Example text**

Product The product (or intersection) of two classes C and D is the class defined by: “… is a member of C and is a member of D”. This may be represented as follows: Figure 11 C ∩ D =df the class defined by: [x ε C ⋅ x ε D] Sum The sum of two class C and D is the class defined by: “… is a member of C or of D”. This may be represented as follows: Figure 12 C ∪ D =df the class defined by: [x ε C v x ε D] Manifold Sum The sum of the elements (or manifold sum) of a class C of classes is the class of members of members of that class.

K-Tuplets With ordered pairs available in the system, the concept of a k-tuplet is introduced in terms of ordered pair as, a set formed out of the sets a1,…, ak (to be called its members) in the following way: A set b is itself a 1-tuplet formed out of b. A (k+1)-tuplet formed out of the sets b1,…, bk+1 is a pair {rs} such that r is a p-tuplet, s is a q-tuplet, and p + q is the same as k + 1. Thus a 2-tuplet formed out of b and c is simply the ordered pair {bc}. It should be noticed that any (k+1)-tuplet is also a k-tuplet, but that its members as a k-tuplet differ from its members as a (k+1)-tuplet.

Three further related concepts are required in the system. These are: Schema of a k-tuplet. Normal k-tuplet. Correspondence between a k-tuplet and a system of values of given variables. The definitions of these are as follows: (1) The schema of a k-tuplet is the result of replacing its members in order by variables b1,…, bk. Thus the schema of the k-tuplet: is: {{{cd}g}h} {{{b1b2}b3}b4} (2) We define a k-tuplet as normal in terms of the degree of each of its members, and the latter is defined in terms of the schema of a k-tuplet as follows: The degree of a variable in the schema of a k-tuplet is the number of pairs of brackets enclosing it; the degree of a member of a k-tuplet is the degree of the corresponding variable in the schema.