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What is most unique about set theory is that it is the perfect amalgam of the visual and the abstract. The notions of set theory, and the ideas behind many of the proofs, present themselves to the inner eye in vivid detail. These pictures are not as overtly visual as those of geometry or calculus. You don’t see
We begin these notes on set theory with a de nition of a set, and the basic notation we use to represent sets. De nition 1. A set X is a collection of elements from a known universe . We have seen sets crop up here and there before.
For any formula Φ(x, f, y, ~w) of the language of set theory, we denote by REC(Φ, N, ~w) the class. {hx, yi : (∃n ∈ N)(∃f)[f : n → V ∧ f(x) = y ∧ ∀m ∈ n Φ(m, f|m, f(m), ~w)]}. We will show, under the appropriate hypothesis, that REC(Φ, N, ~w) is the unique function on N which satisfies the procedure given by Φ.
17 Σεπ 2022 · A set is a collection of things called elements. For example {1, 2, 3, 8} would be a set consisting of the elements 1,2,3, and 8. To indicate that 3 is an element of {1, 2, 3, 8}, it is customary to write 3 ∈ {1, 2, 3, 8}.
Most of the set – theoretic notation is extremely standard, and we shall also employ some frequently used conventions for using “ blackboard bold letters ” and other characters to denote familiar sets and number systems :
Set notation. Sets in general will be denoted here by capital letters: S, T, . . . ; in this book they will almost always be sets of numbers, or toward the end, of points in the plane, i.e., ordered pairs of numbers (x, y). The notation a ∈ S means “the number a is an element of the set S”, or said more simply, “a is in S”.
in set A, or set B, or both sets A and B o. Cardinal number of the union of two finite sets. n(A ∪ B) = n(A) + n(B) – n(A ∩ B) Example: A = (1, 2}, B = {2, 3, 4} ∪ = { , , , } Universal set: set that contains all the elements being considered.
