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Sets Basics and its Representation

Sets

The word set is known to carry the same operation and the meaning as the words collection, class and aggregate. We make no attempt to define it. However the set may be thought of the well defined lists or the collection of the material objects such as the books pens or conceptual objects such as the numbers or the points. Each object of the set is called the elements or the member of the sets.
The collection of the vowels a, e , I, o and u of the English alphabet constitute a set. Here the each of the vowel is an element or the member of the set.

Notation

Sets are usually denoted by the capital letters such as A, B, C, D, E, F, ……..X, Y and the elements of the sets by the small letter such as a, b, c, ..x, y, z.
When we talk about the set we always have to be sure whether an object is an element of or is the member of. If the elements belongs to a specific set it is denoted by, the symbol epsilon, \(\in \) is used to denote "belongs to" or "the element of". Whereas the not epsilon\(\notin\) is used to denote doesn't belong to, or is not the element of. The arrow symbol is used to denote implies.
Example: If A is the set of first natural number. Then 1 belongs to A but 0 doesn't belongs to A.

Specification of the set

A set can be specified or described in several ways. But only the following two ways are used.

  1. Tabular form
    In this method, the elements are listed with out repetition, separate the elements by the commas and encloses them in the braces{}.This method is known as the roster method or listing or the tabular form.
    Example: A = {a, e, i, o, u} is an example of the set in the tabular form.
  2. Set builders form
    Sometimes the specification of the set by a tabular form may be inconvenient or even impossible. So in the such situation, we specify the set by stating the property which an element of the set satisfies. Thus A is the set of the element satisfying the property p then
    A = {x:x satisfies the P}
    Here, x represents the arbitrary element of the set A.
    Example: A = {x: x is a vowel}
Special kind of the sets
There are different kinds of the sets; some of the special kinds of the sets are as follows:
  1. Empty set
    A set having the no element is called the empty set or null set or the void set. It is denoted by the Greek letter phi \(\phi \)
    Example: M = { x: x is a male student in a girl campus } is an empty set.
  2. Finite set
    A set containing the finite number of the elements is called as a finite element.
    A = {x: x is a month of the year} and B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} are the examples of the finite sets.
  3. Infinite set
    A set is not finite, is known as a finite set.
    Example of the infinite set, A: {x: x is an integer}; B= {x: x is a point in the line}
Relations between the sets

A set may have one or the more elements common with another set. Also two set may not have the elements common to them. Depending upon the various possibilities, we have the following relations between the sets.

  1. Subset
    A set A is said to be subset of the set B if the every element of A is also the element of the set B. This relation is denoted by A \(\subseteq \) B. This read as A is contained in B or B contains A.here B is also known as the super set of A as and we write it as B \(\supseteq\) .
    Symbolically, A\( \subseteq\) is defined as x \(\in\) A \(\Rightarrow\) x \(\in\) B.
    If the every element of set A is also the element of set B , but there is at least one element of B which is not the element of set A, then A is also known as the proper subset of B.
    This relation is denoted by A \(\subset\) B .
    A = {x : x is a letter in English alphabet}
    B ={x: x is a vowel}
    Then, B \(\subset\) A.
  2. Equal set
    Two set A and B are said to be equal or identical or same if they have the same elements. They are denoted by A = b.
    Thus if A \(\subseteq\) b then B\(\subseteq A \) , then A = B.
    Also if x \(\in\) a \( \Rightarrow\) x \(\in B \) and x\( \in\) B \(\Rightarrow \) x \(\in\) A , then A = B.
References
SHRESTHA, R.M. n.d.
shrestha, R.M. An introduction to basic mathematics. bhotahity: sukunda pustak bhawan, 2010.

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