Injective Function

Definition

An injective function, or one-to-one function, is a mapping between two sets where every element in the first set maps to a unique element in the second set. This means that no two different elements in the first set can map to the same element in the second set, preserving distinctness throughout the mapping. Understanding injective functions is crucial as they relate to function properties, set mappings, and help determine cardinalities of sets.

5 Must Know Facts For Your Next Test

1. An injective function does not necessarily need to cover all elements of the second set; it just needs to ensure that each output is unique to its input.
2. To test if a function is injective, you can use the horizontal line test on its graph: if any horizontal line intersects the graph more than once, the function is not injective.
3. Injective functions can be defined even if they are not defined over all integers; they can exist between any two sets regardless of size.
4. The inverse of an injective function exists and is also a function, which means that each output can be traced back to a single input.
5. In terms of cardinality, if there is an injective function from set A to set B, then the cardinality of A cannot exceed that of B.

Review Questions

• How can you determine if a function is injective using its graphical representation?
  • To determine if a function is injective from its graph, you can use the horizontal line test. If you can draw any horizontal line that intersects the graph at more than one point, then the function fails to be injective. This indicates that there are at least two different inputs that produce the same output, violating the definition of an injective function.
• What implications does an injective function have for comparing the cardinalities of two sets?
  • When there is an injective function from one set A to another set B, it implies that the cardinality of A is less than or equal to that of B. This means that A cannot have more elements than B since each element of A can be paired uniquely with an element in B. However, B might still contain additional elements that are not mapped from A.
• Evaluate the role of injective functions in establishing relationships between different types of functions and their properties.
  • Injective functions play a critical role in understanding the relationships among various types of functions such as surjective and bijective functions. By defining how inputs relate uniquely to outputs, injectivity helps establish foundational properties required for more complex mathematical concepts. For instance, if a function is known to be injective, we can conclude its inverse will also be a well-defined function. This interconnection aids in exploring deeper implications in areas like combinatorics and analysis regarding countability and mappings between infinite sets.

Related terms

Surjective Function: A surjective function is a mapping where every element in the second set has at least one element from the first set that maps to it, covering all elements in the codomain.

Bijective Function: A bijective function is a function that is both injective and surjective, meaning there is a perfect one-to-one correspondence between elements of the two sets.

Cardinality: Cardinality refers to the number of elements in a set, which helps in comparing sizes of different sets and understanding concepts like injectiveness and surjectiveness.