Introductory Real Analysis
Set Theory
Sets and Functions
Problem
Prove that if $A\cup B = A$ and $A \cap B = A$, then $A = B$.
Solution
Problem
Show that in general $(A-B)\cup B \neq A$.
Solution
Problem
Let $A = {2,4,...,2n,...}$ and $B = {3,6,...,3n,...}$. Find $A\cap B$ and $A-B$
Solution
Problem
Prove that
()
$(A-B)\cap C = (A\cap C) - (B\cap C)$
Solution ():
()
$A\Delta B = (A\cup B) - (A\cap B)$
Solution ():
Problem
Prove that
\[\bigcup_\alpha A_\alpha - \bigcup_\alpha B_\alpha \subset \bigcup (A_\alpha - B_\alpha)\]
Solution
Problem
Let $A_n$ be the set of all positive integers divisible by $n$. Find the sets
()
\[\bigcup_{n=2}^\infty A_n\]
Solution ():
()
\[\bigcap_{n=2}^\infty A_n\]
Solution ():
Problem
Find
()
\[\bigcup_{n=1}^\infty [a+\frac{1}{n}, b-\frac{1}{n}]\]
Solution ():
()
\[\bigcap_{n=1}^\infty [a-\frac{1}{n}, b+\frac{1}{n}]\]
Solution ():
Problem
Let $A_\alpha$ be the set of points lying on the curve \[y = \frac{1}{x^\alpha}\quad(0<x<\infty).\] What is \(\bigcap_{\alpha\geq 1} A_\alpha\)?
Solution
Problem
Let $y = f(x) = <x>$ for all real x, where $<x>$ is the fractional part of $x$. Prove that every closed interval of length 1 has the same image under $f$. What is this image? Is $f$ one-to-one? What is the preimage of the interval $\frac{1}{4}\leq y\leq\frac{3}{4}$? Partition the real line into classes of points with the same image.
Solution
Problem
Given a set $M$, let $\mathfrak{R}$ be the set of all ordered pairs on the form $(a,a)$ with $a\in M$, and let $a R b$ if and only if $(a,b)\in\mathfrak{R}$. Interpret the relation $R$.
Solution
Problem
Give an example of a binary relation which is
()
Reflexive and symmetric, but not transitive
Solution ():
()
Reflexive, but neither symmetric nor transitive
Solution ():
()
Symmetric, but neither reflexive nor transitive
Solution ():
()
Transitive, but neither reflexive nor symmetric
Solution ():
Equivalence of Sets. The Power of a Set
Ordered Sets and Ordinal Numbers
Systems of Sets
Metric Spaces
Basic Concepts
Convergence. Open and Closed Sets
Complete Metric Spaces
Contraction Mappings
Topological Spaces