Chapter 3
A Primer of Real Analysis · 6 exercises
Problem 1
Define a relation on sets by setting \(A \sim B\) if and only if \(|A|=|B| .\) Show that this relation is an equivalence relation.
3 step solution
Problem 2
Let \(I=\\{x: x \in \mathbb{R}, 0 \leq x<1\\}\). Show that
a. \(\quad|I|=|\\{x: x \in \mathbb{R}, 0 \leq x \leq 1\\}|\)
b. \(|I|=|\\{x: x \in \mathbb{R}, 0
6 step solution
Problem 2
Let \(A\) be the set of even integers. Show that \(|A|=\aleph_{0}\).
5 step solution
Problem 3
Let \(I=\\{x: x \in \mathbb{R}, 0 \leq x<1\\}\) and suppose \(a\) and \(b\) are real
numbers with \(a
4 step solution
Problem 3
Verify each of the following: a. If \(A\) is a nonempty subset of \(\mathbb{Z}^{+},\) then \(A\) is either finite or countable. b. If \(A\) is a nonempty subset of a countable set \(B,\) then \(A\) is either finite or countable.
7 step solution
Problem 4
Suppose \(A\) is uncountable and \(B \subset A\) is countable. Show that \(A \backslash B\) is uncountable.
5 step solution