Types of Mapping in Relation and function Class-12। Mapping Class-12

Types of Mapping in Relation to Functions
1. **Injective (One-to-One) Mapping**:
- **Description**: A function \( f: A \to B \) is injective if different elements in the domain \( A \) map to different elements in the codomain \( B \). This means that if \( f(a_1) = f(a_2) \), then \( a_1 = a_2 \).
- **Function Property**: Each element of the domain is mapped to a unique element of the codomain.
- **Example**: \( f(x) = 2x \) is injective because no two different inputs \( x \) produce the same output.
2. **Surjective (Onto) Mapping**:
- **Description**: A function \( f: A \to B \) is surjective if every element in the codomain \( B \) is the image of at least one element from the domain \( A \). This means for every \( b \in B \), there exists at least one \( a \in A \) such that \( f(a) = b \).
- **Function Property**: The function covers the entire codomain.
- **Example**: \( f(x) = x^3 \) is surjective if considered from \( \mathbb{R} \) to \( \mathbb{R} \), because every real number has a real cube root.
3. **Bijective (One-to-One Correspondence) Mapping**:
- **Description**: A function \( f: A \to B \) is bijective if it is both injective and surjective. This means each element in the domain \( A \) maps to a unique element in the codomain \( B \), and every element in \( B \) is mapped by some element in \( A \).
- **Function Property**: The function establishes a perfect pairing between the domain and codomain.
- **Example**: \( f(x) = x + 1 \) from the set of all integers \( \mathbb{Z} \) to \( \mathbb{Z} \) is bijective.
4. **Identity Mapping**:
- **Description**: A function \( f: A \to A \) is an identity mapping if every element maps to itself. That is, \( f(a) = a \) for all \( a \in A \).
- **Function Property**: The function does nothing to the elements of the set; each element remains unchanged.
- **Example**: \( f(x) = x \) for any set \( A \).
5. **Constant Mapping**:
- **Description**: A function \( f: A \to B \) is a constant mapping if all elements in the domain \( A \) map to the same single element in the codomain \( B \). That is, \( f(a) = c \) for some constant \( c \in B \) and all \( a \in A \).
- **Function Property**: The function assigns the same value to all elements of the domain.
- **Example**: \( f(x) = 5 \) for any input \( x \) is a constant function.
6. **Linear Mapping**:
- **Description**: A function \( f: A \to B \) is linear if it satisfies the properties of additivity and scalar multiplication: \( f(x + y) = f(x) + f(y) \) and \( f(ax) = af(x) \) for all \( x, y \in A \) and scalar \( a \).
- **Function Property**: The function preserves the operations of vector addition and scalar multiplication.
- **Example**: \( f(x) = 2x \) is a linear function.
7. **Nonlinear Mapping**:
- **Description**: A function \( f: A \to B \) is nonlinear if it does not satisfy the properties of linearity.
- **Function Property**: The function does not preserve the operations of vector addition and scalar multiplication.
- **Example**: \( f(x) = x^2 \) is a nonlinear function.
These types of mappings describe various ways in which elements from one set (the domain) can be related to elements of another set (the codomain) through functions.
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