5.2 Injective, Surjective, and Bijective Functions
5 min read•august 13, 2024
Functions are like matchmakers between sets. Injective functions ensure each element in the target set has at most one admirer. Surjective functions make sure everyone in the target set gets some love. Bijective functions create perfect pairs.
These special functions help us understand relationships between sets. They're crucial for solving equations, comparing set sizes, and creating inverses. Mastering them unlocks deeper insights into mathematical structures and their connections.
Injective, Surjective, and Bijective Functions
Defining Function Types
Top images from around the web for Defining Function Types
For every b∈B, there is at most one a∈A such that f(a)=b
Each element in the is mapped to by at most one element in the
Example: f(x)=2x from R to R is injective because each output corresponds to a unique input
A function f:A→B is surjective ()
For every b∈B, there is at least one a∈A such that f(a)=b
Each element in the codomain is mapped to by at least one element in the domain
Example: f(x)=x2 from R to [0,∞) is surjective because every non-negative real number has at least one corresponding input
A function f:A→B is bijective
It is both injective and surjective
Each element in the codomain is mapped to by exactly one element in the domain
Example: f(x)=x3 from R to R is bijective because it is both injective and surjective
Function Type Combinations
A function can be:
Injective but not surjective
Example: f(x)=ex from R to (0,∞)
Surjective but not injective
Example: f(x)=x2 from R to [0,∞)
Neither injective nor surjective
Example: f(x)=x2 from [0,∞) to R
Both injective and surjective (bijective)
Example: f(x)=x+1 from R to R
Proving Function Properties
Proving Injectivity
To prove that a function f:A→B is injective
Assume f(a1)=f(a2) for some a1,a2∈A
Show that a1=a2
Demonstrates that each element in the codomain is mapped to by at most one element in the domain
Counterexamples can disprove injectivity
Find elements that violate the definition
Example: For f(x)=x2 from R to [0,∞), f(−1)=f(1)=1, so f is not injective
Proving Surjectivity
To prove that a function f:A→B is surjective
Consider an arbitrary element b∈B
Show that there exists an a∈A such that f(a)=b
Demonstrates that each element in the codomain is mapped to by at least one element in the domain
Counterexamples can disprove surjectivity
Find elements in the codomain that are not mapped to by any element in the domain
Example: For f(x)=x2 from R to R, there is no x∈R such that f(x)=−1, so f is not surjective
Proving Bijectivity
To prove that a function f:A→B is bijective
Prove that it is both injective and surjective using the above techniques
For specific types of functions, algebraic techniques can be used to prove or disprove injectivity and surjectivity
Linear functions: Injectivity and surjectivity depend on the rank of the matrix representation
Polynomial functions: Injectivity can be determined by the degree and leading coefficient, while surjectivity depends on the domain and codomain
Exponential functions: Always injective, and surjective depending on the codomain
Inverse Functions and Injectivity
Conditions for Inverse Functions
A function f:A→B has an inverse if and only if it is bijective (both injective and surjective)
If a function is injective but not surjective
It has a left inverse but not a right inverse
The left inverse is a function g:B→A such that g(f(a))=a for all a∈A
If a function is surjective but not injective
It has a right inverse but not a left inverse
The right inverse is a function h:B→A such that f(h(b))=b for all b∈B
If a function is neither injective nor surjective, it does not have an inverse
Examples of Inverse Functions
The function f(x)=2x+1 from R to R is bijective, so it has an inverse
The is f−1(x)=2x−1
The function f(x)=x2 from [0,∞) to [0,∞) is bijective, so it has an inverse
The inverse function is f−1(x)=x
The function f(x)=ex from R to (0,∞) is injective but not surjective, so it has a left inverse but not a right inverse
The left inverse is f−1(x)=lnx
Applying Function Properties to Problems
Determining Function Properties
Use the definitions of injectivity, surjectivity, and bijectivity to determine whether a given function possesses these properties
Example: Determine whether f(x)=x3−1 from R to R is injective, surjective, or bijective
Injective: Yes, because if [f(a) = f(b)](https://www.fiveableKeyTerm:f(a)_=_f(b)), then a3−1=b3−1, which implies a=b
Surjective: Yes, because for any y∈R, there exists an x=3y+1∈R such that f(x)=y
Bijective: Yes, because it is both injective and surjective
Apply the conditions for the existence of an inverse to determine whether a function has an inverse and, if so, find the inverse function
Example: Determine whether f(x)=2x−3 from R to R has an inverse, and if so, find the inverse function
f is bijective because it is a linear function with a non-zero slope, so it has an inverse
The inverse function is f−1(x)=2x+3
Solving Problems with Function Properties
Use the properties of injective, surjective, and bijective functions to solve problems related to function composition
The composition of two injective functions is injective
The composition of two surjective functions is surjective
The composition of two bijective functions is bijective
Example: If f:A→B and g:B→C are bijective functions, prove that (g∘f)−1=f−1∘g−1
Utilize the concepts of injectivity and surjectivity to solve problems involving cardinality and the sizes of sets
An from A to B implies that ∣A∣≤∣B∣
A from A to B implies that ∣A∣≥∣B∣
Example: If there exists an injective function from set A to set B and a surjective function from set B to set C, prove that ∣A∣≤∣C∣
Apply the properties of injective, surjective, and bijective functions to solve problems in various mathematical contexts
Group theory: Isomorphisms between groups are bijective homomorphisms
Ring theory: Isomorphisms between rings are bijective ring homomorphisms
Linear algebra: Invertible linear transformations are bijective
Example: Prove that if f:G→H is a group isomorphism, then f−1:H→G is also a group isomorphism
Key Terms to Review (19)
Arrow diagram: An arrow diagram is a visual representation that depicts the relationship between elements of two sets, commonly used to illustrate functions. Each element from the first set, known as the domain, is connected to an element in the second set, known as the codomain, through arrows, indicating how each input relates to its corresponding output. This clear graphical method helps in understanding the properties of functions such as injectivity, surjectivity, and bijectivity.
Bijective Function: A bijective function is a special type of function that is both injective (one-to-one) and surjective (onto), meaning that every element in the codomain is mapped to by exactly one element in the domain. This ensures that there is a perfect pairing between the domain and codomain, allowing for the existence of an inverse function. Bijective functions are crucial in various mathematical contexts, such as when discussing the characteristics of continuous functions and homeomorphisms, as well as in understanding the concept of inverse functions.
Cantor-Bernstein-Schoenfeld Theorem: The Cantor-Bernstein-Schoenfeld Theorem states that if there exist injective functions between two sets such that one can map the first set into the second and vice versa, then there exists a bijection between the two sets. This theorem highlights an important relationship between the cardinalities of sets, particularly emphasizing that different sets can have the same size in terms of their number of elements, even when they are infinite. It is especially significant in understanding how injective and surjective functions interact with set sizes.
Codomain: The codomain of a function is the set that contains all possible output values, or the 'target' set that a function maps into. While the codomain defines the range of potential outputs, it does not restrict the actual outputs produced by the function, which are known as the range. Understanding codomain is essential for distinguishing between different types of functions and their properties, particularly in exploring concepts like injective, surjective, and bijective functions.
Completeness of mapping: Completeness of mapping refers to the property of a function or mapping that ensures every element in the codomain has a corresponding element in the domain. This characteristic is crucial for understanding surjective functions, where every output is accounted for, highlighting the relationship between input and output sets. It emphasizes the importance of covering all elements in the target set, distinguishing between mappings that leave elements unmapped and those that achieve full coverage.
Constant function: A constant function is a type of function where the output value remains the same regardless of the input value. This means that no matter what value you put into the function, the result will always be the same fixed number, making it a unique case among different function types. Constant functions can be represented graphically as horizontal lines, and they play an essential role in understanding more complex functions and mappings.
Distinct outputs: Distinct outputs refer to the unique values produced by a function for different inputs. In the context of functions, having distinct outputs means that no two different inputs yield the same output value, which is a defining characteristic of certain types of functions. This concept is crucial in understanding injective functions, where each element in the domain maps to a unique element in the codomain, ensuring that all outputs are distinct.
Domain: In mathematics, the domain of a function refers to the complete set of possible input values (or 'x' values) for which the function is defined. Understanding the domain is crucial because it helps identify which values can be used without causing undefined behavior, such as division by zero or taking the square root of a negative number. The domain also influences the overall behavior and characteristics of functions, including injectivity, surjectivity, and their compositions.
F: a → b: The notation f: a → b represents a function f that maps elements from set a (the domain) to elements in set b (the codomain). This concept is fundamental in understanding how inputs are transformed into outputs and is critical for examining properties like injectivity, surjectivity, and bijectivity, as well as the structure of algebraic systems through homomorphisms and isomorphisms.
F(a) = f(b): The expression f(a) = f(b) indicates that two different inputs, a and b, produce the same output when passed through a function f. This relationship is crucial for understanding the concepts of injective, surjective, and bijective functions, as it helps determine whether a function has unique mappings or if different inputs can share outputs. In essence, this expression raises questions about the behavior of the function and its characteristics in terms of one-to-one correspondence.
Function graph: A function graph is a visual representation of a function that shows the relationship between the input values (often represented on the x-axis) and the output values (represented on the y-axis). This graph allows us to easily observe properties such as injectivity, surjectivity, and bijectivity by analyzing how points on the graph relate to one another. Understanding function graphs is crucial for recognizing the nature of different types of functions and how they map inputs to outputs.
Identity function: The identity function is a function that always returns the same value as its input, essentially acting as a 'do nothing' transformation. This fundamental concept connects to various features of functions, including how it serves as a baseline for other types of functions, such as injective and surjective, and plays a crucial role in the composition of functions. In topology, the identity function helps in understanding continuity and homeomorphisms.
Injective Function: An injective function, or one-to-one function, is a type of function where every element in the domain maps to a unique element in the codomain. This means that no two different inputs can produce the same output, ensuring that each output is associated with only one input. Understanding injective functions helps to analyze relationships between sets and contributes to the larger concepts of functions such as surjectivity and bijectivity.
Inverse Function: An inverse function reverses the mapping of a given function, taking the output values back to their corresponding input values. If a function 'f' maps an element 'x' to 'y', then its inverse 'f^{-1}' will map 'y' back to 'x'. This relationship is key in understanding how functions operate, particularly in identifying whether a function can be inverted and how this relates to the concepts of injectivity and surjectivity.
Inverse Function Theorem: The Inverse Function Theorem is a fundamental result in calculus that provides conditions under which a function has a continuous inverse. It states that if a function is continuously differentiable and its derivative is non-zero at a point, then there exists a neighborhood around that point where the function is one-to-one and has a continuous inverse. This theorem connects to the ideas of injective, surjective, and bijective functions, as well as the concept of inverse functions.
One-to-one: A function is considered one-to-one, or injective, if it assigns distinct outputs to distinct inputs. This means that no two different elements in the domain map to the same element in the codomain. Being one-to-one is crucial in understanding the nature of functions as it affects their invertibility and relationship between sets.
Onto: An 'onto' function, also known as a surjective function, is a type of mapping where every element in the target set (codomain) is covered by at least one element from the domain. This means that for a function to be considered onto, there cannot be any leftover elements in the codomain that are not associated with elements from the domain. Understanding onto functions helps in exploring the relationships between sets and how they map to one another, especially in the context of injective and bijective functions.
Range: The range of a function is the set of all possible output values that result from applying the function to its entire domain. This concept is fundamental because it helps to understand what values can actually be produced by a function and how those outputs relate to the inputs, linking the notion of functions to various types, properties, and transformations.
Surjective Function: A surjective function, also known as an onto function, is a type of function where every element in the codomain has at least one corresponding element in the domain. This means that the function covers the entire codomain, ensuring that no elements are left out. In terms of relationships between sets, a surjective function emphasizes the idea of mapping all outputs while still allowing for multiple inputs to produce the same output.