A. Prove that for all positive natural numbers n, 13 + 23 + ... + n3 = n2⋅(n+1)2/4.
A. Prove that for all positive natural numbers n,
13 + 23 + ... + n3 = n2⋅(n+1)2/4.
B. For any natural number n, let Pn be the statement that n3 + n + 1 is even.
i. Show that for every natural number n ∈ ℕ, we have that Pn ⇒ Pn+1.
ii. Show that Pn is always false.
What is going on here, and how does this relate to induction?
C. Using the ordered pairs definition of the nonnegative rationals ℚ ≥0, verify that the distributive property holds for ℚ ≥0. (You may use the properties of the naturals ℕ, including distributivity, as stated in the notes.)
D. We extend from ℚ≥0 to ℚ by another application of the method of ordered pairs, exactly as we did to extend from ℕ to ℤ. Thus, elements of ℚ can be written as ordered pairs of ordered pairs - we might for example write 1/2 as ((1,2),(0,1)). Explain how to construct addition + on the rational numbers in terms of the coordinates in these ordered pairs.
E. Is it possible to find a group operation ⊕ on a set with 0 elements? With 1 element? Explain why or why not!
F. Give the Dedekind cuts in ℝ ≥0 corresponding to the following. Your definition should not refer to the elements themselves.
i. √10 ii. 3√2 iii. 3√2 + √3
G. Verify that the product of two Dedekind cuts in ℝ ≥0 (as on p17 of the notes) is a Dedekind cut.
H. Prove that F2 (as defined on p20 of the notes) is a field.
I. Find (with proof) the infimums of the sets S = {1 / n : n ∈ ℕ} and T = { (-1)n / n : n ∈ ℕ}. Is either infimum a minimum?
J. Verify that the embedding of ℚ ≥0 in ℝ ≥0 respects ≤. That is, verify for a, b ∈ ℚ ≥0 that if a ≤ b as rational numbers, then also a ≤ b as real numbers.
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