# Basic Concepts of Mathematics (The Zakon Series on by Elias Zakon

By Elias Zakon

This publication is helping the coed whole the transition from simply manipulative to rigorous arithmetic, with issues that conceal easy set thought, fields (with emphasis at the actual numbers), a evaluate of the geometry of 3 dimensions, and houses of linear areas.

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Verify that g ◦ f = h ◦ f , though g = h. ] 13. An equilateral triangle ABC (see FigB ure 7) is carried into itself by these rigid motions: clockwise rotations about its center through 0◦ , 120◦ , and 240◦ (call C′ A′ them r0 , r1 , r2 ) and reflections in its altitudes AA′ , BB ′ , CC ′ (call these reflections ha , hb , hc , respectively). , compute their Figure 7 mutual composites). , r1 r2 = r0 (= the identity map); r1 ha = hc ; ha r1 = hb , etc. ) The maps r0 , r1 , r2 , ha , hb , hc are called the symmetries of the triangle.

Iv) (x, y) + (0, 0) ≡ (x, y); (x, y) · (1, 0) ≡ (x, y). (v) (x, y) + (y, x) ≡ (0, 0). ) (vi) (x, y) · {(p, q) + (r, s)} ≡ (x, y) · (p, q) + (x, y) · (r, s). (vii) If (p, q) < (r, s) then (p, q) + (x, y) < (r, s) + (x, y). Similarly for multiplication, provided, however, that (0, 0) < (x, y). ); we call the pair (x, y) “negative” in this case. Show that (x, y) < (0, 0) iff −(x, y) > (0, 0). 6. The laws proved in Problems 4 and 5 show that ordered pairs (x, y) in N × N , with inequalities and operations defined as above, “behave” like integers (positive, negative and 0) except that equality “=” is replaced by “≡”.

Call it the “k-subsequence”). Show that if every k-subsequence (k = 1, 2, 3, . . ) has a largest term (call it qk , for a given k), then the original sequence {un } has a nonincreasing subsequence formed from all such qk -terms. [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, .