Mappings §

Mappings, or functions, are a core concept in mathematics, representing the relationship between two sets that associates each element of the first set with exactly one element of the second set. This note explores the different types of mappings, their diagrams, and the Vertical Line Test, offering a clear understanding of how these concepts interlink within mathematical theory.

Types of Mappings §

Mappings can be classified into several types based on the nature of their relationships:

• Injective (One-to-One): A mapping $f:A\to B$ is injective if different elements in $A$ map to different elements in $B$. Formally, if $f(a)=f(b)$, then $a=b$.

• Surjective (Onto): A mapping $f:A\to B$ is surjective if for every element in $B$, there exists at least one element in $A$ that maps to it. Essentially, $f(A)=B$.

• Bijective (One-to-One Correspondence): A mapping is bijective if it is both injective and surjective. This means every element in $B$ is mapped to by exactly one element in $A$, and every element in $A$ maps to a unique element in $B$.

Mapping Diagrams §

Mapping diagrams visually represent the relationship between two sets in a function. These diagrams include arrows indicating how elements from the domain (first set) are associated with elements in the codomain (second set). Diagrams for injective mappings show one-to-one connections without any two arrows pointing to the same element in the codomain. Surjective mappings ensure that every element in the codomain has at least one arrow pointing to it. Bijective mappings combine these traits, demonstrating a perfect pairing between the sets.

The Vertical Line Test §

The Vertical Line Test is a visual method used to determine if a curve in the Cartesian coordinate system represents a function. According to this test, a curve represents a function if and only if no vertical line intersects the curve more than once. This ensures that each input (x-coordinate) corresponds to a single output (y-coordinate), adhering to the definition of a function.

Examples §

• An injective mapping example is the function $f(x)=2x$ over real numbers, where no two different $x$ values produce the same $y$.
• A surjective mapping example is the function $f(x)=x^2$ over the non-negative real numbers, where every non-negative number is the square of some real number.
• A bijective mapping example is the function $f(x)=x+3$ over real numbers, where every real number corresponds uniquely to another real number in the codomain.

Applications §

Understanding different types of mappings is crucial in fields like calculus, where injective functions are necessary for inverse functions to exist, and in algebra, for constructing bijections that demonstrate set equivalence. They also play a pivotal role in computer science, especially in data structure and algorithm design.

Test Questions §

1. Define an injective, surjective, and bijective mapping, and give an example of each.
2. Draw a mapping diagram for a function $f:\{1,2,3\}\to \{a,b,c,d\}$ that is surjective.
3. Explain how the Vertical Line Test determines whether a graph represents a function.

By grasping the intricacies of Mappings, students can better understand the relationships between mathematical sets and apply these concepts to various domains, enhancing their analytical and problem-solving skills.