**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: .

**Theorem 2**:The preimages of the intersection of two sets is the intersection of the preimages of the sets: .

**Theorem 3**: The images of the union of two sets equals the union of the images of the sets: .

* 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 , and , 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 :

.

**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 .
- 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 and , then .

Problem 2: Show that in general

Problem 3: Let and . Find and .

Problem 4: Prove that

(a)

(b)

Problem 5: Prove that .

Problem 6: Let be the set of all positive integers divisible by n. Find the sets (a) (b) .

Problem 7: Find

(a) (b)

Problem 8: Let be the set of points lying on the curve where . What is ?

Problem 9: Let for all real x, where 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 ? Partition the real line into classes of points with the same image.

Problem 10: Given a set M, let be the set of all ordered pairs on the form with , and let aRb iff . 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.