Start Learning Sets 5 | Range, Image and Preimage [dark version]

A map’s “range” tells which outputs get hit at all, but the more precise tools are the image of a subset and the pre-image of a subset—definitions that let mathematicians track what happens to selected inputs (or selected outputs) under a function. Starting from the basic setup of a function as a relation that assigns each element of a domain set A to an element of a codomain set B, the discussion first uses a familiar example: sending a natural number x to x². Visualizing arrows from natural numbers to integers makes it clear that not every integer appears as a square, motivating the range as the set of all codomain elements y for which some x in A satisfies f(x)=y.

The range is then generalized in a way that leads naturally to the next step: instead of asking which outputs occur overall, ask which outputs occur from a chosen subset of inputs. For a subset à ⊆ A, the image f(Ã) is defined as the collection of all y in B such that there exists an x in à with f(x)=y. This turns “apply the function” into a set operation: feed in a whole group of inputs, and collect every output they produce. The transcript emphasizes a compact notation often used in textbooks—writing sets like {f(x) | x ∈ Ã}—and warns that parentheses can be confusing when the objects involved are sets rather than single elements.

To show how geometry can guide intuition, the discussion then considers a function on pairs of real numbers: (x1, x2) ↦ x1² + x2². In the plane R×R, points at a fixed distance from the origin form circles, and the formula matches the Pythagorean theorem: x1² + x2² equals the squared radius. That means every point on a circle maps to the same nonnegative number, and varying the radius produces exactly all real values ≥ 0. So the range of this map is the half-line [0, ∞).

The final key concept flips the direction. Given a subset B̃ ⊆ B, the pre-image f⁻¹(B̃) is the set of all x in A that land inside B̃ after applying f—formally, all x such that f(x) ∈ B̃. A crucial clarification follows: the “−1” here does not mean an inverse function; it only signals that the input set is taken from the codomain side.

An explicit number example ties everything together. Define f on natural numbers to integers by mapping even x to 0 and odd x to x itself. The image of {2,3,4} is {0,3} because 2 and 4 are even (both map to 0) while 3 is odd (maps to 3). The pre-image of {0} consists of all even natural numbers, since every even input maps to 0. Together, range, image, and pre-image provide a precise language for tracking how functions move information between sets—either from inputs to outputs, or from outputs back to the inputs that produce them.