By Elias Zakon

This publication is helping the scholar whole the transition from basically manipulative to rigorous arithmetic, with issues that conceal uncomplicated set concept, fields (with emphasis at the genuine numbers), a evaluate of the geometry of 3 dimensions, and houses of linear areas.

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**Example text**

In Example (1) above, un = 2n. Since the domain of a sequence is known to consist of integers, we often omit it and give only the range Du′ , specifying the terms un in the order of their indices n. Thus, instead of (1), we briefly write 2, 4, 6, . . , 2n, . . or, more generally, u1 , u2 , . . , un , . . , along with the still shorter notation {un }. Nevertheless, whatever the notation, the sequence u (a set of ordered pairs) should not be confused with Du′ (the set of single terms un ). A sequence need not be a one-to-one mapping; it may have equal (“repeating”) terms: um = un (m = n).

In what case do these formulas hold with “f −1 ” replaced by “f ”? In what case are they true for both f and f −1 ? 8. , prove that (i) f −1 Ai = f −1 [Ai ]; (ii) f −1 Ai = f −1 [Ai ]. 9. If f is a mapping, show that f [f −1 [A]] ⊆ A and that if A ⊆ Df′ , then f [f −1 [A]] = A. In what case do we have f −1 [f [A]] = A? Give a proof. 10. Which (if any) of the relations ⊆ and ⊇ holds between the sets f [A] ∩ B and f [A ∩ f −1 [B]]? Give a proof. 11. The characteristic function CA of a set A in a space S is defined on S by setting CA (x) = 1 if x ∈ A, and CA (x) = 0 if x ∈ / A.

Hint: Show that qk ≥ qk+1 , k = 1, 2, . . , the maximum term qk cannot increase as the number k of the dropped terms increases. ] ∗ 9. From Problems 7 and 8 infer that every infinite sequence of real numbers {un } has an infinite monotonic subsequence. ] 10. , with domain {1, 2, . . , p}, can one form, given that the range of the sequences is a fixed set of m elements? 11. Let {An } be an infinite sequence of sets. For each n, let n Bn = n Ak , k=1 Cn = ∞ Ak , k=1 Dn = ∞ Ak , k=n En = Ak . k=n Show that the sequences {Bn } and {Dn } are expanding, while {Cn } and {En } are contracting.