Introduction to Set

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Set and Notation

Set theory is a branch of mathematics. It studies the properties of collection of well-defined objects.
A German mathematician Georg Cantor (1845–1918) has conceptualized the modern study of set theory. According to him set is a collection of well-defined objects in which it is possible to determine if a given object is included in the collection.
We use \( \in \) and \( \notin \) symbols to represent if an element is included to a set or not respectively.
For example, with respect to the sets A = {1,2,3,4}, B = {blue, white, red}, and C = {n2 − 4 : n is an integer; and 0 ≤ n ≤ 19} , we can write
4 ∈ A and 285 ∈ C; but
9 ∉ C and green ∉ B.

Set is denoted by a single capital letter (upper cases) of English alphabets such as A, B, C … and so forth. For example
A = {a, e, i, o, u} :The set of vowels

The objects in a set are known as elements or members of the set. The elements in a set are enclosed within middle brackets. For instance, a set A of vowels in English is written within {…} and it is written as
A = {a, e, i, o, u}.

The elements of set are denoted by small letters of English alphabets unless and otherwise stated if applicable. However, the elements in a set can be material objects such as books, pens, people etc. or conceptual objects such as numbers,alphabets, points etc.
Please lease be careful to distinguish the symbols \( 0, \{0\}, \phi, \{\phi\} \)

Describing a Set

When we specify elements of a set, we are simply describing the set. The most common methods used to describe sets are given below.

  1. Semantic definition/ Intentional definition/ Verbal description

    It is also called intentional definition, using a rule or semantic description. This is called verbal description method. For Example

    1. A is the set of first four positive integers.
    2. B is the set of colors used in Nepali flag.
  2. Syntatic definition/ Extensional definition / Listing method

    This method is called roster notation or listing method done by listing each member of the set. This extensional definition is denoted by enclosing the list of members in curly brackets. For Example

    1. C = {4, 2, 1, 3}
    2. D = {blue, white, red}

    In this method, every element of a set must be unique; no two members may be identical. However, the order in which the elements of a set are listed is ignored (unlike for a sequence or tuple).
    Combining these two ideas into an example
    A={6, 11} , B= {11, 6}
    In the examples above, A = B.

  3. Venn-Diagram

    In this method, a diagram is used to represent the relationship among the sets, called Venn-diagram. It was named after an English philosopher John Venn (1834-1923). In a Venn-Diagram, rectangular region represent universal set, and other subsets usually by circular regions (sometimes, it can be expressed by ellipse, sphere etc). For Example
    If U={1,2,3,4,5,6,7,8,9,10}
    A={1,2,3,4,5}
    B={4,5,6,7,8}
    Then, the Venn-Diagram can be described as below.

  4. The set-builder notation
    1. A={x:x is the set of first four positive integers}
    2. B={x|x is the set of colors used in Nepali flag}

Relation Between Sets

In mathematics, a relation between sets is a subset of their Cartesian product. However, we can also define relation as follows.

  1. Subset

    A set \( A\) is said to be a subset of another set \( B\) if every element of \( A\) is also an element of the set \( B\) . If \( A\) is subset of \( B\) , then it is written as \(A \subset B\) and read as “\( A\) is contained in \( B\) ” or “\( A\) is a subset of \( B\) ”.
    If set A is NOT subset of set B, then it is written as \(A \not\subset B\).
    For example:
    Let \(A = \{1, 2, 3\}, B = \{3, 4, 5,6\}\) and \(C = \{1, 2, 3, 4, 5\}\) then \(A \subset C\) but \(B \not\subset C\).
    In usual notation of set of numbers, the relation between them are as below
    \(\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset\mathbb{Q} \subset \mathbb{R} \).
    There are two types of subsets

    1. Proper subset
      If \( A \subset B\) and \( A \ne B\) then \( A\) is called a proper subset of \( B\) . In this case, \( B\) is superset of \( A\)
    2. Improper subset
      If \( A \subset B\) and \( A = B\) then \( A\) is called an improper subset of \( B\) . It is written as \( A \subseteq B\) .
      Note:
      • The empty set \( \phi \) is a subset of every set.
      • Every set is a subset of itself.
      • Every non-empty set has at least two subsets
      • The total number of possible subsets of a set with n-elements is \( 2^n\) .
  2. Power Set

    Let \( S\) is a set. Then the set of all the possible subsets of \( S\) is called power set of \( S\) . It is denoted by \( P(S)\) . For example,
    if \( S = \{a, b, c\}\) then \( P(S) = \{\phi, \{a\}, \{b\}, \{c\}, \{a, b\}, \{b, c\},\{a, c\}, \{a, b, c\}\}\) .

    Thus, the power set of a set S is the set of all subsets of S, including S itself and the empty set. For example, the power set of the set {1, 2, 3} is
    {{1, 2,3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}.
    The power set of a set S usually written as P(S).

    The power set of a finite set with n elements has 2n elements.
    This relationship is one of the reasons for the terminology power set. For example, the set {1, 2, 3} contains three elements, and the power set shown above contains
    23 = 8 elements.
    The power set of an infinite (either countable or uncountable) set is always uncountable.

    • If \( P (S)\) is the power set of a set \( S\) then \( n (P(S)) = 2^{n(S)}\) .
    • Power set of a finite set is finite.
    • \( S \in P(S)\) , that is, Set \( S\) is an element of power set of \( S\) .
  3. Equal Sets

    Two or more sets are called equal (or identical or same) if they consist same elements. For example,
    if \( A = \{1, 2\}\) and \( B = \{2, 1\}\) then \( A = B\) but the sets \( C = \{12\}\) and \( D =\{21\}\) are not equal.

  4. Equivalent Sets

    Two sets \( A\) and \( B\) are called equivalent if their cardinal number is same, i.e., \( n (A) = n (B)\) . The symbol to denote equivalent sets is “\( \sim\) ”. For example,
    if \( A = \{1, 2, 3\}\) and \( B = \{p, q, r\}\) then \( A \sim B\).

  5. Overlapping Sets

    Two sets \( A\) and \( B\) are called overlapping set if they do have some common element. For example,
    if \( A = \{1,2,3\}\) and \( B = \{3,4,5\}\) then \( A\) and \( B\) are overlapping sets as \( \{3\}\) is common to both sets \( A\) and \( B\) .

  6. Disjoint Sets

    Two sets \( A\) and \( B\) are called disjoint if they have no elements in common. For example, if \( A = \{1, 2, 3\}\) and \( B = \{4,5, 6\}\) then \( A\) and \( B\) are disjoint as they have no element in common.

  7. Comparable Sets

    Two sets \( A\) and \( B\) are said to be comparable if \( A \subset B \) or \( B \subset A\) . For example, the sets \( A=\{a, b, c\}\) , and \( C=\{a, b, c, d\}\) are comparable. But the sets \( C=\{a, b, c\}\) and \( D=\{a, c, d, e\}\) are not comparable (incomparable) sets because C and D are overlapping sets.

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