Set Theory for Real Analysis: Part I:

Reference: Introductory Real Analysis by Kolmogorov and Fomin:

(Available at, for example, Amazon India: http://www.amazon.in/Introductory-Analysis-Dover-Books-Mathematics/dp/0486612260/ref=sr_1_1?s=books&ie=UTF8&qid=1499886166&sr=1-1&keywords=Introductory+real+analysis)

Functions and mappings. Images and preimages.

Theorem 1: The preimage of the union of two sets is the union of the preimages of the sets: $f^{-1}(A \bigcup B) = f^{-1}(A)\bigcup f^{-1}(B)$ .

Theorem 2:The preimages of the intersection of two sets is the intersection of the preimages of the sets: $f^{-1}(A \bigcap B)=f^{-1}(A) \bigcap f^{-1}(B)$ .

Theorem 3: The images of the union of two sets equals the union of the images of the sets: $f(A \bigcup B)=f(A) \bigcup (B)$ .

Remark 1: Surprisingly enough, the image of the intersection of the two sets does not necessarily equal the intersection of the images of the sets. For example, suppose the mapping f projects the xy-plane onto the x-axis, carrying the point (x,y) into the (x,0). Then, the segments $0\leq x\leq 1$ , $y=0$ and $0\leq x \leq 1$ , $y=1$ do not intersect, although their images coincide.

Remark 2: Theorems 1-3 (above) continue to hold for unions and intersections of of an arbitrary number (finite or infinite) of sets $A_{\alpha}$ :

$f^{-1}(\bigcup_{\alpha}A_{\alpha})=\bigcup_{\alpha}f^{-1}(A_{\alpha})$

$f^{-1}(\bigcap_{\alpha}A_{\alpha})=\bigcap_{\alpha}f^{-1}(A_{\alpha})$

$f(\bigcup_{\alpha}A_{\alpha})=\bigcup_{\alpha}f(A_{\alpha})$ .

Decomposition of a set into classes. Equivalence relation.

(NP: This is, of course, well-known so I will not dwell on it too much nor reproduce too much from the mentioned text). Just for quick review purposes:

A relation R on a set M is called an equivalence relation (on M) if it satisfies the following three conditions:

Reflexivity: aRa, for every $a \in M$ .
Symmetry: If aRb, then bRa.
Transitivity: If aRb and bRc, then aRc.

Theorem 4: A set M can be partitioned into classes by a relation R (acting as a criterion for assigning two elements to the same class) if and only if R is an equivalence relation on M.

Remark: Because of theorem 4, one often talks about the decomposition of M into equivalence classes.

Exercises 1:

Problem 1: Prove that if $A \bigcup B=A$ and $A \bigcap B=A$ , then $A=B$ .

Problem 2: Show that in general $(A-B)\bigcup B \neq A$

Problem 3: Let $A = \{ 2,4, \ldots, 2n, \ldots\}$ and $B= \{ 3,6, \ldots, 3n, \ldots\}$ . Find $A \bigcap B$ and $A-B$ .

Problem 4: Prove that

(a) $(A-B)\bigcap C=(A \bigcap C)-(B \bigcap C)$

(b) $A \triangle B =(A \bigcup B)-(A \bigcap B)$

Problem 5: Prove that $\bigcup_{\alpha}A_{\alpha}-\bigcup_{\alpha}B_{\alpha} \subset \bigcup_{\alpha}(A_{\alpha}-B_{\alpha})$ .

Problem 6: Let $A_{\alpha}$ be the set of all positive integers divisible by n. Find the sets (a) $\bigcup_{n=2}^{\infty}A_{\alpha}$ (b) $\bigcap_{n=2}^{\infty}A_{\alpha}$ .

Problem 7: Find

(a) $\bigcup_{n=1}^{\infty}[a+\frac{1}{n},b-\frac{1}{n}]$ (b) $\bigcap_{n=1}^{\infty}(a-\frac{1}{n},b+\frac{1}{n})$

Problem 8: Let $A_{\alpha}$ be the set of points lying on the curve $y=\frac{1}{x^{\alpha}}$ where $(0<x<\infty)$ . What is $\bigcap_{\alpha \geq 1}A_{\alpha}$ ?

Problem 9: 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 the image? is f one-to-one? What is the pre-image of the interval $\frac{1}{4} \leq y \leq \frac{3}{4}$ ? Partition the real line into classes of points with the same image.

Problem 10: Given a set M, let $\Re$ be the set of all ordered pairs on the form $(a,a)$ with $a\in M$ , and let aRb iff $(a,b) \in \Re$ . Interpret the relation R.

Problem 11: Give an example of a binary relation which is:

a) Reflexive and symmetric, but not transitive.

b) Reflexive but neither symmetric nor transitive.

c) Symmetric, but neither reflexive nor transitive.

d) Transitive, but neither reflexive not symmetric.

MScMathematics

Set Theory for Real Analysis: Part I:

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