IIT JEE Main Maths -Sets and their representation - Study Notes-New Syllabus
IIT JEE Main Maths -Sets and their representation – Study Notes – New syllabus
IIT JEE Main Maths -Sets and their representation – Study Notes -IIT JEE Main Maths – per latest Syllabus.
Key Concepts:
- Sets and Their Representation
Sets and Their Representation
A set is a well-defined collection of distinct objects. These objects are called the elements or members of the set.
Notation: Sets are usually denoted by capital letters \( A, B, C, \dots \) and elements by small letters \( a, bFF, c, \dots \). If an element \( a \) belongs to a set \( A \), we write \( a \in A \); otherwise, \( a \notin A \).
Methods of Representing Sets:
- Roster or Tabular Form: Elements are listed, separated by commas, and enclosed in curly brackets. Example: \( A = \{0,1, 2, 3, 4, 5\} \)
- Set-builder Form: Describes the property of the elements. Example: \( A = \{x \mid x \text{ is a whole number less than } 8\} \)
Types of Sets:
| Type | Description | Example |
|---|---|---|
| Empty Set | Contains no elements | \( \varnothing \text{ or } \{\} \) |
| Finite Set | Contains a finite number of elements | \( A = \{1, 2, 3\} \) |
| Infinite Set | Contains infinitely many elements | \( N = \{1, 2, 3, \dots\} \) |
| Equal Sets | Contain exactly the same elements | \( A = \{1, 2\}, B = \{2, 1\} \) |
| Equivalent Sets | Contain the same number of elements | \( A = \{a, b, c\}, B = \{1, 2, 3\} \) |
Important Symbols:
- \( \in \): belongs to (membership)
- \( \notin \): does not belong to
- \( \subset \): subset
- \( \subseteq \): subset or equal
- \( \supset \): superset
- \( n(A) \): number of elements in set \( A \)
Subset:
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 written as \( A \subseteq B \).
If \( A \subseteq B \) and \( A \ne B \), then \( A \) is called a proper subset of \( B \), denoted \( A \subset B \).
Example
Let \( A = \{2, 4\} \) and \( B = \{1, 2, 3, 4, 5\} \). Determine whether \( A \) is a subset or proper subset of \( B \).
▶️ Answer / Explanation
All elements of \( A \) (i.e., 2 and 4) are also elements of \( B \). Therefore, \( A \subseteq B \).
Since \( A \ne B \), \( A \) is also a proper subset: \( A \subset B \).
Example
Let \( A = \{1, 2, 3, 4, 5\} \) and \( B = \{x \mid x \text{ is an even number less than } 6\} \). Represent both sets using roster form and determine whether they are equal or equivalent.
▶️ Answer / Explanation
Step 1: Write set \( B \) in roster form:
\( B = \{2, 4\} \)
Step 2: Compare with set \( A \).
Set \( A = \{1, 2, 3, 4, 5\} \) has 5 elements, while \( B \) has 2 elements.
Step 3: Since the elements are not the same, \( A \ne B \).
However, both are finite sets, so we can check if they are equivalent.
Step 4: The number of elements:
\( n(A) = 5, \quad n(B) = 2 \)
Conclusion: \( A \) and \( B \) are neither equal nor equivalent.
Example
Express the set \( A = \{2, 4, 6, 8, 10\} \) in set-builder form, and explain what property defines its elements.
▶️ Answer / Explanation
Step 1: Observe the pattern in the elements of \( A \).
The numbers are even natural numbers less than or equal to 10.
Step 2: Express in set-builder form.
\( A = \{x \mid x \text{ is an even natural number, } 2 \le x \le 10\} \)
Alternate form: \( A = \{x \mid x = 2n, \, n \in \mathbb{N}, \, n \le 5\} \)
Step 3: Verify correspondence:
If \( n = 1, 2, 3, 4, 5 \), then \( x = 2, 4, 6, 8, 10 \), which matches the given set.
Conclusion: The set-builder form \( A = \{x \mid x = 2n, \, n \in \mathbb{N}, \, n \le 5\} \) correctly represents the given set.
Notes and Study Materials
- Concepts of Sets and Relations
- Sets and Relations Master File
- Sets and Relations Revision Notes
- Sets and Relations Formulae
- Sets and Relations Reference Book
- Sets and Relations Past Many Years Questions and Answer
Examples and Exercise
IIT JEE (Main) Mathematics ,”Sets and Relations” Notes ,Test Papers, Sample Papers, Past Years Papers , NCERT , S. L. Loney and Hall & Knight Solutions and Help from Ex- IITian
About this unit
Sets and their representation. Union, intersection, and complement of sets and their algebraic properties. Power set. Relation, Types of relations, equivalence relations.
IITian Academy Notes for IIT JEE (Main) Mathematics – Sets and Relations
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