SETS

The concept of a set is one of the most fundamental in mathematics. One defines the term set as ” a well-defined collection of objects. The objects that make up a set (also known as the set’s elements or members) can be anything: numbers, people, letters of the alphabet, other sets, and so on.

We assume that the word ” set” is synonymous with the words “collection”, ” aggregate”, ” class” and is comprised of elements. The words “element”, ” object”, ” member, are synonymous.

If $a$  is an element of a set $A$, then we write $a \in A$  and, say $a$  belongs to $A$  or $a$  is in $A$  or $a$  is a member of $A$. If $a$  does not belong to $A$, then we write $a \notin A$. It is assumed here that if $A$  is any set and $a$  is any element, then either $a \in A$  or $A \notin A$  and the two possibilities are mutually exclusive.

The following are some illustrations of sets.

• The collection of vowels in English alphabets. This set contains five elements, namely $a, e, i, o, u$.
• The collection of first five prime natural numbers is a set containing the elements $2, 3, 5,7, 11$.
• The collection of all states in the Indian Union is a set.
• The collection of past presidents of the Indian Union is a set.
• The collection of cricketers in the world who were out for 99 runs in a test match is a set.
• The collection of good cricket players of India is not a set, Since the term “good player” is vague and not well defined

In this chapter we will have frequent interaction with some sets, so we reserve some letters for these sets as listed below

• $N$  : for the set of natural numbers
• $Z$  : for the set of integers
• $Z^+$  : for the set of all positive integers
• $Q$  : for the set of all rational numbers
• $Q^+$  : for the set of all positive rational numbers
• $R$  : for the set of all real numbers
• $R^+$  : for the set of all positive real numbers
• $C$ : for the set of all complex numbers

DESCRIPTION OF A SET

A set is often described in the following two forms i.e (i) Roster form or Tabular form (ii) Set-builder form

ROSTER FORM: In this form a set is described by listing elements, separated by commas, within braces $\{ \ \}$.

For example: The set of vowels of English Alphabet may be described as $\{a, e, i, o, u\}$.

Note:

• The order in which the elements are written in a set makes no difference. Thus, $\{a, e, i, o, u\}$ and $\{e,a,i,o,u\}$ denote the same set.
• Also, the repetition of an element has no effect. For example, $\{1,2, 3,2\}$ is the same set as $\{1,2, 3\}$.

SET BUILDER FORM: In this form, a set is described by a characterizing property $P(x)$ of its elements $x$. In such a case the set is described by $\{x : P(x) \ holds \}$  or, $\{x | P(x) \ holds\}$, which is read as ‘the set of all $x$ such that $P(x)$  holds’. The symbol $'|'$  or $':'$  is read as ‘such that’.

The set $E$  of all even natural numbers can be written as $E = \{x : x \in N, x = 2n , n \in N\}$  or  $E = \{x \in N : x = 2n , n \in N\}$

For example:

• The set $A = \{1, 2, 3, 4, 5, 6, 7, 8\}$ can be written as $A = \{x \in N : x \leq 8\}$
• The set of all real numbers greater than $-1$ and less than $1$ can be described as $\{x \in R:-1< x < 1\}$

TYPES OF SETS

EMPTY SET: A set is said to be empty or null or void set if it has no element and it is denoted by $\phi$.  In the roster method $\phi$ is denoted by $\{ \ \}$.

The empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

It follows from this definition that a set $A$ is an empty set if the statement $x \in A$ is not true for any $x$.

For example: $\{x \in R : x^2 = -1 \}= \phi \ or \ \{x \in N : 5 < x < 6\} = \phi$

A set consisting of at least one element is called a non-empty or non-void set.

Note:

• If $A$ and $B$ are any two empty sets then $x \in A$ if and only if $x \in B$ is satisfied because there are no elements $x$ in either $A$ or $B$ to which the condition may be applied. Thus, $A = B$. Hence, there is only one empty set and we denote it by $\phi$.

SINGLETON SET: A set consisting of a single element is called a singleton set.

For example: The set $\{5\}$  is a singleton set. Also $\{x : x \in N \ and \ x^2=49\}$ is a singleton set $\{7\}$

FINITE SET: A set is called a finite set if it is either void set or its elements can be listed (counted, labelled) by natural numbers $1, 2, 3, ...$, and the process of listing terminates at a certain natural number $n$ (say).

In other words, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting.

CARDINAL NUMBER OF A FINITE SET: The number of distinct elements in a finite set is called its cardinal number. It is denoted as $n(A)$ and read as ‘the number of elements of the set’. For example: (i) Set $A = \{ 2, 4, 5, 9, 15 \}$ has $5$ elements.

INFINITE SET: An infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Some examples are: the set of all integers, $\{ \cdots , -1, 0, 1, 2, \cdots \}$, is a countably infinite set; and the set of all real numbers is an uncountably infinite set.

EQUIVALENT SETS: Two finite sets $A$ and $B$ are equivalent if their cardinal numbers are same. i.e. $n(A) = n(B)$.

EQUAL SETS: Two sets $A$ and $B$ are said to be equal if every element of $A$ is a member of $B$, and every element of $B$ is a member of $A$.

If sets $A$ and $B$ are equal, we write $A = B$ and $A \neq B$ when $A$ and $B$ are not equal.

It follows from the above definition and the definition of equivalent sets that equal sets are equivalent but equivalent sets need not be equal. For example, $A = \{1,2, 3\}$ and $B = \{a, b, c\}$ are equivalent sets but not equal sets.

SUBSETS

Let $A$ and $B$ be two sets. If every element of $A$ is an element of $B$, then $A$ is called a subset of $B$.

If $A$ is a subset of $B$, we write  $A \subseteq B$ which is read as $'A'$ is a subset of $'B'$ or ‘$'A'$ is contained in, $'B'$ .

Thus, $A \subseteq B$ if and only if $a \in A \Rightarrow a \in B$.

If $A$ is a subset of $B$, we say that $B$ contains $A$ or, $B$ is a super set of $A$ and we write $B \supset A$.

If $A$ is not a subset of $B$, we write  $A \nsubseteq B$.

Obviously, every set is a subset of itself and the empty set is subset of every set.

Note: Satisfies transitivity i.e. $A \subseteq B$ and $B \subseteq C \Rightarrow A \subseteq C$

A proper subset contains some but not all of the elements of the original set. $A$ is called a proper subset of $B$ If $A \subseteq B$ and $A \neq B$ then $A \subset B$. In such a case, we also say that $B \supset A$. Thus, if $A$ is a proper subset of $B$, then there exists an element $x \in B$ such that $x \notin A$.

An improper subset is a subset containing every element of the original set.

The empty set is a proper subset of a given set.

It follows immediately from this definition and the definition of equal sets that two sets $A$ and $B$ are equal if and only if $A \subseteq B$ and $B \subseteq A$.

Thus, whenever it is to be proved that two sets $A$ and $B$ are equal, we must prove that $A \subseteq B$ and $B \subseteq A$.

SOME RESULTS ON SUBSETS

THEOREM 1: Every set is a subset of itself.

Proof: Let, $A$ be any set. Then each element of $A$ is clearly in $A$ itself. Hence, $A \subseteq A$.

THEOREM 2: The empty set is a subset of every set.

Proof: Let $A$ be any set and $\phi$ be the empty set. In order to show that $\phi \subseteq A$, we must show that every element of $\phi$ is an element of $A$ also. But $\phi$ contains no elements. Hence, every element of $\phi$  is in $A$. Hence $\phi \subseteq A$.

THEOREM 3: The total number of subsets of a finite set containing $n$ elements is $2^n$.

Proof: Let $A$ be a finite set containing $n$ elements. Let $0 \leq r \leq n$. Consider those subsets of $A$ that have $r$ elements each. We know that the number of ways in which $r$ elements can be chosen out of $n$ elements is $^nC_r$. Therefore, the number of subsets of $A$ having $r$ elements each is $^nC_r$. Hence the total number of subsets of $A$ is

$^nC_0 + ^nC_1 + ^nC_2 + \cdots ^nC_n = (1+1)^n = 2^n$

SUBSETS OF THE SET $R$ OF REAL NUMBERS

Following sets are important subsets of the set It of all real numbers:

• The set of all natural numbers $N = \{1, 2, 3, ,4 ,5 ,6 , \cdots\}$
• The set of all integers $Z = \{ \cdots , -3, -2, 1, 0, 1, 2, ,3 ,4, \cdots \}$
• The set of all rational numbers $Q = \{x: x =$ $\frac{m}{n}$ $, m, n \in Z, n \neq 0\}$
• The set of all irrational numbers. It is denoted by $"T"$ Thus, $T = \{x : x \in R \ and \ x \notin Q \}$

Clearly $N \subset Z \subset Q \subset R$. $T \subset R$ and $N \not \subset T$

INTERVALS AS SUBSETS OF $R$

On real line various types of infinite subsets are designated as intervals as defined below:

CLOSED INTERVAL Let $a$ and $b$ be two given real numbers such that $a < b$. Then, the set of all real numbers $x$ such that $a \leq x \leq b$ is called a closed interval and is denoted by $[a, b]$.

Thus $[a, b] = \{x \in R : a \leq x \leq b\}$

OPEN INTERVAL lf $a$ and $b$ are two real numbers such that $a < b$, then the set of all real numbers $x$ satisfying $a is called an open interval and is denoted by $(a, b)$ or $] a, b [$.

Thus $(a, b) = \{ x \in R: a

SEMI-OPEN or SEMI-CLOSED INTERVAL lf $a$ and $b$ are two real numbers such that $a < b$, then the sets $(a, b] = \{ x \in R: a and $[a, b) = \{ x \in R: a \leq x are known as  semi-open and semi closed intervals. $(a, b]$ and $[a, b)$ are also denoted by $]a,b]$ and $[a,b[$ respectively.

The number $b-a$ is called the length of any of the intervals $(a,b),[a,b], [a,b)$ and $(a,b]$.

These notations provide an alternative way of designating the subsets of the set $R$ of all real numbers. For example, the interval $[0, \infty)$ denotes the set $R^+$ of all non-negative real numbers, while the interval $(-\infty, 0)$ denotes the set $R^-$ of all negative real numbers. The interval $(-\infty, \infty)$ denotes the set $R$ of all real numbers.

UNIVERSAL SET

A super set of each of the given sets. Such a set is called the universal set and is denoted by $U$. Thus, a set that contains all sets in a given context is called the universal set.

If $A= \{1,2,3\}, \ B = \{2,4,5,6\}$  and $C = \{1,3, 5,7\}$, then $U = \{1,2, 3,4,5,6,7\}$ can be taken as the universal set.

POWER SET

Let $A$ be a set. Then the collection or family of all subsets of $A$ is called the power set of $A$ and is denoted by $P(A)$.

That is $P(A) = \{ S : S \subset A\}$

Since the empty set and the set $A$ itself are subsets of $A$ and are therefore elements of $P(A)$. Thus, the power set of a given set is always non-empty.

Let $A= \{1,2,3\}$. Then, the subsets of $A$ are $\phi , \{1\}, \{2\}, \{3\}, \{1,2\}, \{1, 3\}, \{2, 3\}$ and $\{1,2,3\}$. Hence, $P(A) = \{ \phi, \{1\}, \{2), \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\} \}$

We know that a set having $n$ elements has $2^n$ subsets. Therefore, if $A$ is a finite set having $n$ elements, then $P(A)$ has $2^n$ elements. The cardinal number of $P(A)$ is $2^n$.

VENN DIAGRAMS

In Venn-diagrams the universal set $U$ is represented by points within a rectangle and its subsets are represented by points in closed curves (usually circles) within the rectangle.

• If a set $A$ is a subset of a set $B$, then the circle representing $A$ is drawn inside the circle representing $B$ as shown in Figure.
• If $A$ and $B$ are not equal but they have some common elements, then to represent $A$ and $B$ we draw two intersecting circles as shown in the figure.
• Two disjoint sets are represented by two non-intersecting circles as shown in the figure.

OPERATIONS ON SETS

UNION OF SETS: Let $A$ and $B$ be two sets. The union of $A$ and $B$ is the set of all those element, which belong either to $A$ or to $B$ or to both $A$ and $B$.

We shall use the notation $A \cup B$ (read as $"A \ union \ B"$) to denote the union of $A$ and $B$.

Thus $A \cup B = \{x: x \in A$ or $x \in B\}$

Clearly $x \in A \cup B \Leftrightarrow x \in A$ or $x \in B$ or $x \in B$. And $x \notin A \cup B \Leftrightarrow x \notin A$ and $x \notin B$

In the figure below, the shaded part represents $A \cup B$. It is evident from the definition that $A \subseteq A \cup B , B \subseteq A \cup B$.

If $A$ and $B$ are two sets such that $A \subset B$, then $A \cup B = B$. Also, $A \cup B = A$, if $B \subset A$.

lf $A = \{1, 2, 3 \}$ and $B = \{1, 3, 5,7 \}$, then $A \cup B = \{ 1, 2, 3, 5,7 \}$

If $A_1, A_2, A_3, \cdots A_n$ is a finite family of sets, then their union is denoted by $\underset{i=1}{\overset{n}{U}}$ or $A_1 \cup A_1 \cup A_1 \cup \cdots \cup A_n$

INTERSECTION OF SETS Let $A$and $B$ be two sets. The intersection of $A$ and $B$ is the set of all those elements that belong to both $A$ and $B$.

Intersection of $A$ and $B$ is denoted by $A \cap B$ (read as $" A \ intersection \ B"$ ).

Thus $A \cap B = \{x:x \in A$ and $x \in B \}$

Clearly, $x \in A \cap B \Leftrightarrow x \in A$ and $x \in B$

In the figure shown, the shaded region is $A \cap B$. Evidently, $A \cap B \subseteq A$ and $A \cap B \subseteq B$

If $A$ and $B$ are two sets, then $A \cap B = A$ if $A \subset B$ and $B \cap A = B$ if $B \subset A$.

If $A_1, A_2, A_3, \cdots A_n$ is a finite family of sets, then their intersection is denoted by $\underset{i=1}{\overset{n}{U}}$ or $A_1 \cap A_1 \cap A_1 \cap \cdots \cap A_n$

If $A= \{1,2,3,1,5,6,7 \}$, $B= \{ 2,4,6,8,10 \}$ and $C= \{ 4,6,7,8,9,10, 11 \}$, then $A \cap B = \{ 2,4,6 \}$. Therefore $A \cap B \cap C = \{4, 6\}$

DISJOINT SETS Two sets $A$ and $B$ are said to be disjoint, if $A \cap B = \phi$. If $A \cap B \neq \phi$ then $A$ and $B$ are said to be’intersecting or overlapping sets.

DIFFERENCE OF SETS Let $A$ and $B$ be two sets. The difference of $A$ and $B$, written as $A - B$, is the set of all those elements of $A$ which do not belong to $B$.

Thus $A - B = \{ x: x \in A$  and $x \notin B \}$  or $A - B = \{ x \in A : x \notin B \}$.

Clearly, $x \in A - B \Leftrightarrow x \in A$ and $x \notin B$

Similarly,the difference $B-A$ is the set of all those elements of $B$ that do not belong to $A$ i.e  $B - A = \{ x \in B : x \notin A \}$

As shown in the figure, the shaded part represents $A - B$ and also $B - A$ .

SYMMETRIC DIFFERENCE OF TWO SETS If $A$ and $B$ are two sets, the symmetric difference of set $A$ and set $B$ is the set $(A-B) \cup (B-A)$ and is denoted by $A \triangle B$.

COMPLEMENT OF A SET Let $U$ be the universal set and let $A$ be a set such that $A \subset U$. Then, the complement of $A$ with respect to $U$ is denoted by $A'$ or $A^c$ or $U - A$ and is defined the set of all those elements of $U$ which are not in $A$.

Thus $A' = \{ x \in U : x \notin A \}$. Clearly, $x \in A \Leftrightarrow x \notin A$

Let the set of natural numbers $N = \{ 1, 2, 3, 4, \cdots \}$ be the universal set and let $A = \{2, 4, 6,8, \cdots \}$. Then $A'=\{ 1,3,5, \cdots \}$.

LAWS OF ALGEBRA OF SETS

THEOREM 1 (Idempotent Laws) For any set $A$ (i) $A \cup A = A$ and (ii) $A \cap A = A$

THEOREM 2 ( Identity Laws) For any set $A$, (i) $A \cup \phi = A$  and (ii) $A \cap U = A$. i.e. $\phi$ and $U$ are identify elements for union and intersection respectively.

THEOREM 3 (Commutative Laws) For any two sets $A$ and $B$ (i) $A \cup B = B \cup A$ and (ii) $A \cap B = B \cap A$ i.e. union and intersection are commutative.

THEOREM 4 (Associative Laws) If $A, B$ and $C$ are three sets, then (i) $(A \cup B) \cup C = A \cup (B \cup C)$ and (ii) $A \cap (B \cap C) = (A \cap B) \cap C$

THEOREM 5 (Distributive Laws) If $A, B$ and $C$ are three sets, then (i) $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ and (ii) $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ i.e. union and intersection are distribution over intersection and union respectively.

THEOREM 6 (De-Morgan’s Laws) If $A$ and $B$ are any two sets, then (i) $(A \cup B)' = A' \cap B'$ and (ii) $(A \cap B)' = A' \cup B'$

MORE RESULTS ON OPERATIONS ON SETS

THEOREM 1 If $A$ and $B$ are any two sets, then

(i) $A - B = A \cap B'$

(ii) $B - A = B \cap A'$

(iii) $A - B = A \Leftrightarrow A \cap B = \phi$

(iv) $(A - B) \cup B = A \cup B$

(v) $(A - B) \cap B = \phi$

(vi) $A \subseteq B \Leftrightarrow B' \subseteq A'$

(vii) $(A-B) \cup (B-A) = (A \cup B) - (A \cap B)$

THEOREM 2 If $A, \ B$ and $C$ are any three sets, then prove that:

(i) $A-(B \cap C)=(A-B) \cup (A-C)$

(ii) $A-(B \cup C)=(A-B) \cap (A-C)$

(iii) $A \cap (B -C) =(A \cap B) -(A \cap C)$

(iv) $A \cap (B \triangle C) =(A \cap B) \triangle (A \cap C)$

SOME IMPORTANT RESULTS ON NUMBER OF ELEMENTS IN SETS
If $A, B$ and $C$ are finite sets, and $U$ be the finite universal set, then

(i) $n (A \cup B) = n(A) + n(B) - n(A \cap B)$ or, $n (A \cap B) = n(A) + n(B) -n(A \cup B)$

(ii) $n(A \cup B) =n(A) + n(B) \Leftrightarrow A, B$ are disjoint non-void sets.

(iii) $n(A -B) =n(A) -n(A \cap B)$ i.e. $n(A -B)+n(A \cap B) =n(A)$

(iv) $n(A \triangle B) =$ No. of elements which belong to exactly one of $A$ or $B$ $= n(A) +n(B) -2n(A \cap B)$

(v) $n(A \cup B \cup C) =n(A) +n(B) +n(C) -n(A \cap B) -n(B \cap C) -n(A \cap C) + n(A \cap B \cap C)$

(vi) Number of elements in exactly two of the sets $A, B, C$
$= n(A \cap B) +n(B \cap C) + n(C \cap A) - 3n(A \cap B \cap C)$

(vii) Number of elements in exactly one of the sets $A, B, C$
$=n(A) +n(B) +n(C) - 2 n(A \cap B) - 2n(B \cap C)-2n(A \cap C) + 3n(A \cap B \cap C)$

(viii) $n(A' \cup B')= n((A \cap B)') = n(U) -n(A \cap B)$

(ix) $n (A' \cap B') = n((A \cup B)' ) =n(U)-n(A \cup B)$