One-to-one

Definition

A function is called one-to-one (or injective) if different inputs map to different outputs, meaning that no two distinct elements in the domain are assigned to the same element in the codomain. This property ensures that each output is uniquely associated with one input, establishing a clear and distinct relationship between the sets involved.

5 Must Know Facts For Your Next Test

1. A function f: A → B is one-to-one if f(a_1) = f(a_2) implies that a_1 = a_2 for any a_1 and a_2 in A.
2. Graphically, a function is one-to-one if any horizontal line drawn intersects the graph at most once.
3. If a function is one-to-one, its inverse function exists and is also a function.
4. In the context of set theory, one-to-one functions help establish relationships between different sets by ensuring unique associations.
5. Not all functions are one-to-one; for example, the function f(x) = x² is not one-to-one because both positive and negative inputs yield the same output.

Review Questions

• How can you determine if a given function is one-to-one using its graph?
  • To determine if a function is one-to-one using its graph, you can apply the horizontal line test. This means drawing horizontal lines across the graph; if any horizontal line intersects the graph at more than one point, the function is not one-to-one. Conversely, if every horizontal line intersects at most once, then the function meets the criteria for being one-to-one.
• What are the implications of a function being one-to-one regarding its inverse?
  • If a function is one-to-one, it guarantees that an inverse function can be defined. This means that for each output in the codomain, there is exactly one corresponding input from the domain. As a result, this relationship allows us to reverse the mapping effectively. If a function does not have this injective property, its inverse would not be a well-defined function because multiple inputs could correspond to the same output.
• Evaluate how one-to-one functions contribute to understanding relationships between sets in mathematics.
  • One-to-one functions play a crucial role in understanding relationships between sets by establishing unique mappings from elements in one set to another. This injective property ensures that every element in the first set corresponds to a distinct element in the second set without overlap. Consequently, this allows mathematicians to analyze these relationships quantitatively and qualitatively, aiding in areas such as cardinality comparison and creating bijections between finite and infinite sets.

Related terms

Injective Function:

A type of function where every element of the range is mapped by at most one element of the domain.

Surjective Function:

A function where every element of the codomain is mapped by at least one element of the domain.

Bijective Function:

A function that is both injective and surjective, creating a perfect one-to-one correspondence between elements of the domain and codomain.