## REAL ANALYSIS BOOK BY SK MAPA

#### Real analysis by sk mapa

Download real analysis by sk mapa.pdf

real analysis by sk mapa.pdf

Download real analysis by sk mapa pdf

real analysis by sk mapa pdf

Set theory is a large and complicated subject in its own right. There is no time

in this course to touch on any but the simplest parts of it. Instead, we’ll just look

at a few topics from what is often called “native set theory,” many of which should

already be familiar to you.

We begin with a few definitions.

A set is a collection of objects called elements. Usually, sets are denoted by the

capital letters A, B, · · · , Z. A set can consist of any type and number of elements.

Even other sets can be elements of a set. The sets dealt with here usually have real

numbers as their elements.

If a is an element of the set A, we write a ∈ A. If a is not an element of the set

A, we write a ∈/ A.

If all the elements of A are also elements of B, then A is a subset of B. In this

case, we write A ⊂ B or B ⊃ A. In particular, notice that whenever A is a set, then

A ⊂ A.

Two sets A and B are equal, if they have the same elements. In this case we write

A = B. It is easy to see that A = B iff A ⊂ B and B ⊂ A. Establishing that both of

these containments are true is the most common way to show two sets are equal.

If A ⊂ B and A 6= B, then A is a proper subset of B. In cases when this is

important, it is written A $ B instead of just A ⊂ B.

There are several ways to describe a set.

A set can be described in words such as “P is the set of all presidents of the

United States.” This is cumbersome for complicated sets.

All the elements of the set could be listed in curly braces as S = {2, 0, a}. If the

set has many elements, this is impractical, or impossible.

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