mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the horizontal line test is a test used to determine whether a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...

(i.e., one-to-one).

In calculus

A ''horizontal line'' is a straight, flat line that goes from left to right. Given a function

f \colon \mathbb \to \mathbb

(i.e. from the

real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

to the real numbers), we can decide if it is

by looking at horizontal lines that intersect the function's

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

. If any horizontal line

y=c

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

y=c

) but different x values, which by definition means the function cannot be injective. Variations of the horizontal line test can be used to determine whether a function is

surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...

bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...

: *The function ''f'' is surjective (i.e., onto)

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

its graph intersects any horizontal line at least once. *''f'' is bijective if and only if any horizontal line will intersect the graph exactly once.

In set theory

Consider a function

f \colon X \to Y

with its corresponding

as a subset of the

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...

X \times Y

. Consider the horizontal lines in

X \times Y

\ = X \times \

. The function ''f'' is

each horizontal line intersects the graph at most once. In this case the graph is said to pass the horizontal line test. If any horizontal line intersects the graph more than once, the function fails the horizontal line test and is not injective.

References

{{reflist Basic concepts in set theory

In calculus

In set theory

See also

References