What is a One-to-One Function? Breaking it Down

What is a One-to-One FunctionOh, mathematics. You either love it or hate it.  But I posit it doesn’t have to be this way.  There are many parts of mathematics that can be learned as easily as one can improve their reading. Too many young students make the decision that because they will never use math later in life they don’t really need to learn it. Not only is that wrong outright, math can lead to promising career options and opportunities down the road. Nonetheless, let’s focus here on breaking down what a one-to-one function is all about.

The One-to-One Function

A one-to-one (injective) function f from set X to set Y is a function such that each x in X is related to a different y in Y. Such a function can be labeled an injective function, injection or a one-to-one function. More formally, a one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.

In other words, every element of the function’s codomain is the image of at most one element of its domain. The term one-to-one function must not to be confused with one-to-one correspondence (aka surjective injection or bijective function), which uniquely maps all elements in both domain and codomain to each other.

Occasionally, an injective function from X to Y is denoted fX ↣ Y, using an arrow with a barbed tail (U+21A3 ↣ rightwards arrow with tail). The set of injective functions from X to Y may be denoted YX using a notation derived from that used for falling factorial powers, since if X and Y are finite sets with respectively m and n elements, the number of injections from X to Y is nm (see the twelvefold way).

A function f that is not injective is sometimes called many-to-one. However, this terminology is also sometimes used to mean “single-valued”, i.e., each argument is mapped to at most one value.

Let f be a function whose domain is a set A. The function f is injective if and only if for all a and b in A, if f(a) = f(b), then a = b; that is, f(a) =f(b) implies a = b.  Equivalently, if a ≠ b, then f(a) ≠ f(b).

The Horizontal Line Test

In mathematics, the horizontal line test is a test used to determine whether a function is a one-to-one function.

According to Wikipedia, a horizontal line is a straight, flat line that goes from left to right. Given a function  (i.e. from the real numbers to the real numbers), we can decide if it is injective by looking at horizontal lines that intersect the function’s graph. If any horizontal line  intersects the graph in more than one point, the function is not injective. To see this, note that the points of intersection have the same y-value (because they lie on the line ) but different x values, which by definition means the function cannot be injective.

Other Important Properties

  • If f and g are both injective, then f o g is injective.
  • If g o f is injective, then f is injective (but g need not be).
  • If f : X → Y is injective if and only if, given any functions gh : W → X, whenever f o g = f o h, then g = h. In other words, injective functions are precisely the monomorphisms in the category set of sets.
  • If f : X → Y is injective and A is a subset of X, then f −1(f(A)) = A. Thus, A can be recovered from its image f(A).
  • If f : X → Y is injective and A and B are both subsets of X, then f(A ∩ B) = f(A) ∩ f(B).
  • Every function h : W → Y can be decomposed as h = f o g for a suitable injection f and surjection g. This decomposition is unique up to isomorphism, and f may be thought of as the inclusion function of the range h(W) of h as a subset of the codomain Y of h.
  • If both X and Y are finite with the same number of elements, then f : X → Y is injective if and only if f is surjective (in which case f is bijective).

Mathematics can scare most but it really doesn’t have to be that way. Learning mathematics from the most basic level to the most advanced level is made easier when utilizing as many resources as possible.  Be sure to enroll in one of our easy-to-learn courses:

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