131 lines
3.9 KiB
Plaintext
131 lines
3.9 KiB
Plaintext
/-
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I 3.12 Exercises
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A Set of Axioms for the Real-Number System
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-/
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-- ========================================
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-- Exercise 1
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--
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-- If `x` and `y` are arbitrary real numbers with `x < y`, prove that there is
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-- at least one real `z` satisfying `x < z < y`.
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-- ========================================
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-- # TODO
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-- ========================================
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-- Exercise 2
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--
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-- If `x` is an arbitrary real number, prove that there are integers `m` and `n`
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-- such that `m < x < n`.
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-- ========================================
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-- # TODO
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-- ========================================
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-- Exercise 3
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--
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-- If `x > 0`, prove that there is a positive integer `n` such that `1 / n < x`.
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-- ========================================
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-- # TODO
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-- ========================================
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-- Exercise 4
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--
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-- If `x` is an arbitrary real number, prove that there is exactly one integer
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-- `n` which satisfies the inequalities `n ≤ x < n + 1`. This `n` is called the
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-- greatest integer in `x` and is denoted by `⌊x⌋`. For example, `⌊5⌋ = 5`,
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-- `⌊5 / 2⌋ = 2`, `⌊-8/3⌋ = -3`.
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-- ========================================
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-- # TODO
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-- ========================================
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-- Exercise 5
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--
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-- If `x` is an arbitrary real number, prove that there is exactly one integer
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-- `n` which satisfies `x ≤ n < x + 1`.
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-- ========================================
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-- # TODO
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-- ========================================
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-- Exercise 6
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--
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-- If `x` and `y` are arbitrary real numbers, `x < y`, prove that there exists
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-- at least one rational number `r` satisfying `x < r < y`, and hence infinitely
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-- many. This property is often described by saying that the rational numbers
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-- are *dense* in the real-number system.
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-- ========================================
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-- # TODO
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-- ========================================
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-- Exercise 7
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--
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-- If `x` is rational, `x ≠ 0`, and `y` irrational, prove that `x + y`, `x - y`,
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-- `xy`, `x / y`, and `y / x` are all irrational.
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-- ========================================
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-- # TODO
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-- ========================================
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-- Exercise 8
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--
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-- Is the sum or product of two irrational numbers always irrational?
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-- ========================================
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-- # TODO
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-- ========================================
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-- Exercise 9
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--
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-- If `x` and `y` are arbitrary real numbers, `x < y`, prove that there exists
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-- at least one irrational number `z` satisfying `x < z < y`, and hence
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-- infinitely many.
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-- ========================================
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-- # TODO
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-- ========================================
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-- Exercise 10
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--
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-- An integer `n` is called *even* if `n = 2m` for some integer `m`, and *odd*
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-- if `n + 1` is even. Prove the following statements:
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--
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-- (a) An integer cannot be both even and odd.
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-- (b) Every integer is either even or odd.
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-- (c) The sum or product of two even integers is even. What can you say about
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-- the sum or product of two odd integers?
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-- (d) If `n²` is even, so is `n`. If `a² = 2b²`, where `a` and `b` are
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-- integers, then both `a` and `b` are even.
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-- (e) Every rational number can be expressed in the form `a / b`, where `a` and
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-- `b` are integers, at least one of which is odd.
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-- ========================================
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-- # TODO
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-- ========================================
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-- Exercise 11
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--
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-- Prove that there is no rational number whose square is `2`.
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--
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-- [Hint: Argue by contradiction. Assume `(a / b)² = 2`, where `a` and `b` are
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-- integers, at least one of which is odd. Use parts of Exercise 10 to deduce a
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-- contradiction.]
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-- ========================================
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-- # TODO
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-- ========================================
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-- Exercise 12
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--
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-- The Archimedean property of the real-number system was deduced as a
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-- consequence of the least-upper-bound axiom. Prove that the set of rational
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-- numbers satisfies the Archimedean property but not he least-upper-bound
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-- property. This shows that the Archimedean property does not imply the
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-- least-upper-bound axiom.
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-- ========================================
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-- # TODO |