By Elias Zakon

This publication is helping the coed whole the transition from basically manipulative to rigorous arithmetic, with themes that disguise easy set concept, fields (with emphasis at the actual numbers), a evaluation of the geometry of 3 dimensions, and houses of linear areas.

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Prove the following: (i) R[A ∪ B] = R[A] ∪ R[B]; (ii) R[A ∩ B] ⊆ R[A] ∩ R[B]; (iii) R[A − B] ⊇ R[A] − R[B]. Generalize formulas (i) and (ii) by proving them with A, B replaced by an arbitrary family of sets {Ai } (i ∈ I). Disprove the reverse inclusions in (ii) and (iii) by counterexamples (thus showing that equality may fail). Also, try to prove them and explain where and why the proof fails. 5. State and prove necessary and sufficient conditions for the following: (i) R[x] = ∅; (ii) R−1 [x] = ∅; (iv) R−1 [A] = ∅.

Therefore we 38 Chapter 1. Some Set Theoretical Notions consider it here, even though it involves the notion of integers, to be formally introduced in Chapter 2, along with real numbers. Definition 1. By an infinite sequence we mean a mapping (call it u) whose domain Du consists of all positive integers 1, 2, 3, . . (it may also contain 0). A finite sequence is a mapping u in which Du consists of positive (or nonnegative) integers less than some fixed integer p. ). Note 1. In a wider sense, one may speak of “sequences” in which Du also contains some negative integers, or excludes some positive integers.

U2k−1 , . . were selected, we would obtain the subsequence 1, 3, 7, 19, 29, 37, . . The first subsequence could briefly be denoted by {u2k } (here nk = 2k); the second subsequence is {u2k−1 }, nk = 2k − 1, k = 1, 2, . . , u1 is the first term, u2 the second , and so on. This procedure is actually well known from everyday life: by numbering the buildings in a street or the books in a library, we put them in a certain order or sequence. The question now arises: given a set A, is it always possible to “number” its elements by integers?