# Set Theory for Real Analysis: Part I:

Reference: Introductory Real Analysis by Kolmogorov and Fomin:

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 2Theorems 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:

1. Reflexivity: aRa, for every $a \in M$.
2. Symmetry: If aRb, then bRa.
3. 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. 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.