Download HW 7

MATH 113, HOMEWORK 7
Due November 1, 2013
The goal of this homework is to construct the real numbers R, an example of the continuum satisfying all of the axioms (including axiom 4). We will use the method of Dedekind
cuts of rational numbers. We assume knowledge of the rational numbers Q and we may
freely use all basic properties of Q. In particular, Q satisfies axioms 1 – 3 of the continuum
and all properties of C that we have proved on sheets 1 and 2 (but NOT sheet 3!).
Definition. A Dedekind cut in Q is a nonempty subset A ⊂ Q satisfying:
(1) If a ∈ A and x ∈ Q satisfies x < a, then x ∈ A.
(2) A has an upper bound: there exists u ∈ Q such that a ≤ u for all a ∈ A.
(3) A does not have a last point.
Let R denote the set of all Dedekind cuts in Q. We call R the set of real numbers. For
A, B ∈ R, define A < B if A ⊂ B and A 6= B.
1. Prove that (R, <) satisfies axioms 1 – 3.
2. Let X ⊂ R be a nonempty set of real numbers that is bounded above. Prove that
sup X exists. (N.B.: by problem 1, you may use properties of the continuum, but not
anything depending on axiom 4, which we have yet to prove for R.)
3. Let X ⊂ R be a nonempty set of real numbers that is bounded below. Prove that
inf X exists.
4. Suppose that A, B ∈ R satisfy A < B. Then there exists some P ∈ R such that
A < P < B.
5. Suppose that P ∈ R is either a least upper bound or greatest lower bound of an open
set X ⊂ R of real numbers. Prove that P is a limit point of both X and R \ X.
6. Suppose that X ⊂ R is a nonempty set of real numbers that is both open and closed
satisfying X 6= R. Derive a contradiction.
7. Prove that R is connected.
8. The definition of Dedekind cuts makes sense with Q replaced by any continuum satisfying axioms 1 – 3. What would R look like if we had used Z instead of Q? What
property of Q that is missing from Z makes our construction of the real numbers
sensible?
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9. It is very satisfying to have an example of the continuum that satisfies axiom 4. But
the real numbers should also behave like numbers! Given two Dedekind cuts A, B ∈ R,
propose a definition of the addition A ⊕ B and the multiplication A B of A and
B. (I suggest that you use the fancy symbols ⊕ and for these operations so that
they do not get confused with the ordinary addition + and multiplication · of rational
numbers.)
10. Let us now check that your definition of the addition and multiplication of real numbers
is sensible.
(i) Define 0 ∈ R and show that it satisfies A ⊕ 0 = A for all A ∈ R.
(ii) Define 1 ∈ R and show that it satisfies A 1 = A for all A ∈ R.
(iii) Explain why multiplication is distributive over addition:
A (B ⊕ B 0 ) = (A B) ⊕ (A B 0 )
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for all A, B, B 0 ∈ R.