Sets: Empty, Finite, Infinite, and Equal Sets Study of Equality and Subsets

The Empty Set

Definition 1: A set which does not contain any element is called the empty set or the null set or the void set.
Notation: The empty set is denoted by the symbol $\phi$ or the empty braces ${}$ .
Criteria for Comparison: * Example: Consider a student studying in both Class X and Class XI simultaneously. Since this is impossible, the set $B$ of such students contains no element. Therefore, $B$ is an empty set. * Example: Let A = \{x : x \text{ is an integer and } -3 \le x < 7\}. This is not an empty set and can be written in roster form as $\{-2, -1, 0, 1, 2, 3, 4, 5, 6\}$ .
Specific Examples of Empty Sets: * (i) A = \{x : 1 < x < 2, x \in N\}. There is no natural number between $1$ and $2$ , so $A$ is empty. * (ii) $B = \{x : x^2 - 2 = 0 \text{ and } x \text{ is a rational number}\}$ . The equation is not satisfied by any rational value, making $B$ an empty set. * (iii) $C = \{x : x \text{ is an even prime number greater than 2}\}$ . Since 2 is the only even prime, $C$ is empty. * (iv) $D = \{x : x^2 = 4, x \text{ is odd}\}$ . The equation $x^2 = 4$ has solutions $2$ and $-2$ , neither of which is odd.

Finite and Infinite Sets

Definition 2: A set which is empty or consists of a definite number of elements is called finite; otherwise, the set is called infinite.
Cardinality Notation: By the number of elements of a set $S$ , we mean the number of distinct elements, denoted by $n(S)$ . If $n(S)$ is a natural number, then $S$ is a non-empty finite set.
Examples of Finite Sets: * $A = \{1, 2, 3, 4, 5\}$ where $n(A) = 5$ . * $B = \{a, b, c, d, e, g\}$ where $n(B) = 6$ . * $C = \{\text{men living presently in different parts of the world}\}$ . Though the number is large, it is a finite natural number. * $W$ = The set of days of the week. * $S$ = The set of solutions of the equation $x^2 - 16 = 0$ .
Examples of Infinite Sets: * The set of natural numbers ( $N$ ). * $G$ = The set of points on a line. * Odd natural numbers ( $1, 3, 5, 7, \dots$ ).
Roster Form for Infinite Sets: Infinite sets are represented by writing a few elements that indicate the structure, followed (or preceded) by three dots ( $\dots$ ). * Integers: $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$ . * Constraint: Not all infinite sets can be described in roster form. For example, the set of real numbers ( $R$ ) cannot be described this way because its elements do not follow a particular pattern.
Exercise Example 6 Analysis: * (i) $\{x : x \in N \text{ and } (x-1)(x-2) = 0\}$ : Finite, elements are $\{1, 2\}$ . * (ii) $\{x : x \in N \text{ and } x^2 = 4\}$ : Finite, elements are $\{2\}$ . * (iii) $\{x : x \in N \text{ and } 2x - 1 = 0\}$ : Finite (empty set, as $x = \frac{1}{2} \notin N$ ). * (iv) $\{x : x \in N \text{ and } x \text{ is prime}\}$ : Infinite. * (v) $\{x : x \in N \text{ and } x \text{ is odd}\}$ : Infinite.

Equal Sets

Definition 3: Two sets $A$ and $B$ are equal ( $A = B$ ) if they have exactly the same elements. If they do not, they are unequal ( $A \neq B$ ).
Rules for Equality: * Repetition of elements does not change a set. Example: $A = \{1, 2, 3\}$ and $B = \{2, 2, 1, 3, 3\}$ are equal because every element of $A$ is in $B$ and vice-versa. * Order of elements does not matter. Example: $A = \{1, 2, 3, 4\}$ and $B = \{3, 1, 4, 2\}$ are equal.
Comparative Examples: * The set of letters in "ALLOY" ( $X = \{A, L, O, Y\}$ ) and the set of letters in "LOYAL" ( $B = \{L, O, Y, A\}$ ) are equal sets. * $A = \{0\}$ is not equal to B = \{x : x > 15 \text{ and } x < 5\} because $B$ is empty ( $\phi$ ). * $C = \{x : x - 5 = 0\}$ contains only $\{5\}$ . $E = \{x : x \text{ is an integral positive root of } x^2 - 2x - 15 = 0\}$ roots are $5$ and $-3$ , but only $5$ is a positive root. Thus, $C = E$ . * $D = \{x : x^2 = 25\}$ contains $\{-5, 5\}$ . Therefore, $C \neq D$ because $-5 \notin C$ .

Subsets

Definition 4: A set $A$ is said to be a subset of a set $B$ if every element of $A$ is also an element of $B$ . This is expressed in symbols as $A \subset B$ .
Logical Notation: $A \subset B \text{ if } a \in A \implies a \in B$ .
Properties of Subsets: * Every set is a subset of itself: $A \subset A$ . * The empty set is a subset of every set: $\phi \subset A$ . * Two-way implication for equality: $A = B$ if and only if (iff) $A \subset B$ and $B \subset A$ . This is written as $A \subset B \text{ and } B \subset A \iff A = B$ .
Example Case: If $X$ is the set of all students in your school and $Y$ is the set of all students in your class, then $Y \subset X$ .

Exercises and Applications

Exercise 1.2 Identification

Null Sets: * Set of odd natural numbers divisible by 2 (Null). * Set of even prime numbers (Not null, contains $\{2\}$ ). * \{x : x \text{ is a natural number, } x < 5 \text{ and } x > 7\} (Null). * $\{y : y \text{ is a point common to any two parallel lines}\}$ (Null).
Finite vs Infinite Identifying: * Months of a year (Finite). * $\{1, 2, 3, \dots\}$ (Infinite). * $\{1, 2, 3, \dots, 99, 100\}$ (Finite). * The set of positive integers greater than 100 (Infinite). * The set of prime numbers less than 99 (Finite). * Lines parallel to the x-axis (Infinite). * Letters in the English alphabet (Finite). * Numbers multiple of 5 (Infinite). * Animals living on the earth (Finite). * Circles passing through the origin (0,0) (Infinite).
Set Equality Comparisons: * $A = \{a, b, c, d\}, B = \{d, c, b, a\}$ : $A = B$ . * $A = \{4, 8, 12, 16\}, B = \{8, 4, 16, 18\}$ : $A \neq B$ (12 is not in B, 18 is not in A). * $A = \{2, 4, 6, 8, 10\}, B = \{x : x \text{ is positive even integer and } x \le 10\}$ : $A = B$ . * $A = \{x : x \text{ is a multiple of 10}\}, B = \{10, 15, 20, 25, 30, \dots\}$ : $A \neq B$ because 15 is not a multiple of 10. * $A = \{2, 3\}, B = \{x : x^2 + 5x + 6 = 0\}$ : $B = \{-2, -3\}$ , so $A \neq B$ . * $A = \{\text{letter in FOLLOW}\} = \{F, O, L, W\}$ , $B = \{\text{letter in WOLF}\} = \{W, O, L, F\}$ : $A = B$ .