Sets and Their Representations

Understanding Sets

• Sets are well-defined collections of objects.

• A collection of objects becomes a set if it is "well-defined."

• To determine if a collection is well-defined, check if the definition is opinion-independent.

• Opinion independence means that the answer to whether an object belongs to the collection should be the same regardless of who provides the answer.

• If the answer varies depending on who is answering, it is not well-defined, and thus not a set.

Examples of Sets

• Example 1: Months starting with 'J' (January, June, July) form a well-defined set because the answer is consistent regardless of who answers.

• Example 2: The 10 most talented writers of India do not form a well-defined set because "most talented" is subjective and varies from person to person.

• Example 3: The 11 best cricket batsmen of the world do not form a well-defined set due to the subjective nature of "best."

• Example 4: All boys in your class form a well-defined set because the members are fixed and do not change based on personal opinions.

• Example 5: Natural numbers less than 100 form a well-defined set since the collection is fixed (1 to 99).

• Example 6: Novels written by Munshi Premchand form a well-defined set because the collection of novels is fixed.

• If the question was changed to "The collection of best novels" then it would no longer be a set due to the subjective assessment by different people.

• Example 7: All even integers form a well-defined set because the starting point and progression are consistent (2, 4, 6, 8, …).

Standard Notations for Sets

• $\mathbb{N}$ represents the set of all natural numbers.

• $\mathbb{Z}$ represents the set of all integers.

• $\mathbb{Q}$ represents the set of all rational numbers.

• $\mathbb{R}$ represents the set of all real numbers.

• $\mathbb{Z}^+$ represents the set of positive integers.

• $\mathbb{Z}^-$ could represent the set of negative integers.

• $\mathbb{Q}^+$ represents the set of positive rational numbers.

• $\mathbb{R}^+$ represents the set of positive real numbers.

• $\mathbb{Q}^-$ could represent the set of negative rational numbers.

• $\mathbb{R}^-$ could represent the set of negative real numbers.

Element Membership in a Set

• Sets are typically represented by uppercase letters (e.g., A).

• Sets are denoted using curly brackets { } enclosing the elements.

• Elements within a set can be numbers, letters, names, or any objects.

• The notation $a \in A$ means that element 'a' belongs to set A.

• The notation $a \notin A$ means that element 'a' does not belong to set A.

Example

Given set $A = {1, 2, 3, 4, 5, 6}$

• $5 \in A$ (5 belongs to A)

• $8 \notin A$ (8 does not belong to A)

• $0 \notin A$ (0 does not belong to A)

• $4 \in A$ (4 belongs to A)

• $2 \in A$ (2 belongs to A)

• $10 \notin A$ (10 does not belong to A)

Representation of Sets

• Two primary methods to represent sets:

  • Roster Form

  • Set-Builder Form

Roster Form

• All elements are listed within curly brackets, separated by commas.

• Elements satisfying a specific condition (stated in the question) are listed.

• Each element is listed only once; repetition is not allowed.

• The order of elements does not matter.

• Example: If a set is represented as {1, 2, 2, 3}, it should be written as {1, 2, 3} because elements should not be repeated inside of sets.

• {1, 2, 3}, {1, 3, 2}, and {3, 2, 1} all refer to the same set.

Set-Builder Form

• Involves specifying a unique condition that the elements must satisfy.

• Uses a symbolic notation to define the set based on a rule or condition.

Examples: Roster Form

Convert the following set-builder notations to roster form:

1. A = {x : x \in \mathbb{Z}, -3 \leq x < 7}
   Solution: $A = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}$

   The set of integers greater than or equal to -3 and strictly less than 7.

2. B = {x : x \in \mathbb{N}, x < 6}
   Solution: $B = {1, 2, 3, 4, 5}$

   The set of natural numbers (positive integers) strictly less than 6.

3. $C = {x : x \in \mathbb{N}, \text{x is a 2-digit number, sum of digits is 8}}$
   Solution: $C = {17, 26, 35, 44, 53, 62, 71, 80}$

   The set of 2-digit natural numbers where the sum of the digits equals 8.

4. $D = {x : x \text{ is a prime number, divisor of 60}}$
   Solution: $D = {2, 3, 5}$

   The set of prime numbers that are divisors of 60.

5. $E = \text{Set of all letters in the word ``TRIGONOMETRY''}$
   Solution: $E = {T, R, I, G, O, N, M, E, Y}$

   The set of unique letters in the word ``TRIGONOMETRY", without repetition.

6. $F = \text{Set of all letters in the word ``BETTER''}$
   Solution: $F = {B, E, T, R}$

   The set of unique letters in the word ``BETTER”, without repetition.

Examples: Set-Builder Form

Convert the following roster form notations to set-builder:

1. ${3, 6, 9, 12}$
   Solution: $A = {x : x = 3n, n \in \mathbb{N}, n \leq 4}$

   The set of multiples of 3, where n is a natural number less than or equal to 4.

2. ${2, 4, 8, 16, 32}$
   Solution: $B = {x : x = 2^n, n \in \mathbb{N}, n \leq 5}$

   The set of powers of 2, where n is a natural number less than or equal to 5.
   Another way to write this is: $B = {x : x = 2^n, n \in \mathbb{N}, 1 \leq n \leq 5}$*

3. ${5, 25, 125, 625}$
   Solution: $C = {x : x = 5^n, n \in \mathbb{N}, n \leq 4}$

   The set of powers of 5, where n is a natural number less than or equal to 4.

4. ${2, 4, 6, …}$
   Solution:

   • Using language: $D = {x : x \text{ is an even natural number}}$

   • Using mathematical notation: $D = {x : x = 2n, n \in \mathbb{N}}$

   The set of even natural numbers or multiples of 2.
   *Note: Even numbers can be represented as $2n$, and odd numbers can be represented as $2n + 1$.

5. ${1, 4, 9, …, 100}$
   Solution: $E = {x : x = n^2, n \in \mathbb{N}, n \leq 10}$

   The set of squares where n is a natural number less than or equal to 10.