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.
Problem 1: Prove that if and , then .
Problem 2: Show that in general
Problem 3: Let and . Find and .
Problem 4: Prove that
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
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.