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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