Function
In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y.
At a glance
Key signals
0% cross fields · reaches 5 more
- Mathematics
- Algebra
- Set Theory
- Explanation
- Examples
- Misconception
- Sourced relations 6
- Attribution
Dependencies
What this concept builds on and what it makes possible — derived from the atlas’s dependency, causal and structural relations, not from every related edge.
Foundations · builds on
Cartesian coordinate systemenablesFunctionEstablished
Open Cartesian coordinate system →
Cartesian coordinate system enables function.
Mechanism: Coordinates let a function be drawn as a curve, turning an abstract rule into a visible shape.
- Wikipedia (English & German editions) verifiedmoderate evidence
Functiondepends onSetEstablished
function depends on set.
Mechanism: A function depends on sets: it is a rule assigning to each element of one set (the input) exactly one element of another (the output).
- Wikipedia (English & German editions) verifiedmoderate evidence
Functiondepends onVariableEstablished
function depends on variable.
Mechanism: A function assigns each input value of a variable exactly one output — a rule of dependence between quantities.
- Wikipedia (English & German editions) verifiedmoderate evidence
Enables · leads to
Axiom of choicedepends onFunctionEstablished
Axiom of Choice depends on Function.
System context
Functionis aRelationEstablished
Function is a kind of Relation.
Continuous functionis aFunctionEstablished
continuous function is a kind of function.
Mechanism: A continuous function is a function whose graph has no breaks or jumps — small changes in input give small changes in output.
- Wikipedia (English & German editions) verifiedmoderate evidence
Sine and cosineis aFunctionEstablished
sine and cosine is a kind of function.
Mechanism: Sine and cosine are functions of an angle: they return the coordinates of a point moving around the unit circle.
- Wikipedia (English & German editions) verifiedmoderate evidence
Structural role & consequence
Interpreted from the current atlas graph — what the connections mean, not just how many there are.
Currently dark in the atlas: no key date stored · 12 of 12 of its relations lack claim-level evidence.
atlas representation · Describes the current Thinking OS representation, not the state of the world.
Builds on 3 foundations (requires / depends-on / derived-from / emerges-from).
structural · Structural graph analysis — not a claim of importance, causation or history.
Structural neighbourhood: 12 → 40 → 97 concepts reachable within 3 hops.
structural · Structural reach — being reachable is not the same as being understood.
All 12 of its relationships stay within its own discipline — a field-specific concept in the current atlas.
structural · Structural graph analysis — not a claim of importance, causation or history.
cross-field
12 within-field, 0 cross-field
Strengths & constraints
Constraints
- Evidence coverage currently thin in the atlas — few of its relationships carry claim-level evidence. atlas representation
- No dated history stored — the atlas records no key date for this concept. atlas representation
Conditions
- Its dependency reading rests on 3 foundation relations. structural
- Read structurally — most of its relationships carry no external evidence yet, so claims here are graph-derived. structural
Dependency radial
What this concept builds on (left) and what it makes possible (right) — derived from dependency and causal relations.
What builds on this
1 concept build on this directly, 1 in total, across 2 disciplines.
Structural downstream reach along dependency edges — not a claim of historical necessity.
Seen through each discipline
How this concept sits in each of its fields — derived from its real connections in the graph, not asserted.
Through this lens it connects to Set, Morphism, Fourier analysis and Dynamical systems.
Through this lens it connects to Variable and Function composition.
Through this lens it connects to Set, Axiom of choice and Relation.
Check yourself
A quick check against a common misconception. Nothing is scored — picking the tempting-but-wrong answer just flags an idea worth revisiting.
Which statement is correct?
A function can give several outputs for one input.
By definition each input has exactly one output. A rule that returns two values for one input (like ±√x) is a relation, not a function.
Look for: Learner accepts a graph that fails the vertical-line test as a function.
Related ideas to explore
Concepts that look related but are not yet connected here — candidates for a connection to reason about, not established links.
This idea also appears in…
The same structure shows up in other disciplines. These are real recurrences drawn from the graph — a starting point for asking “what carries over, and what changes?”
Cause and effect32 disciplines · 30 concepts
Concepts
shares a mental model
shares a mental model
shares a mental model
shares a mental model
shares a mental model
shares a mental model
A function is a rule that takes an input and gives exactly one output. Put a number in, get a number out — like a machine where the same input always produces the same result.
A function maps each element of an input set (the domain) to exactly one element of an output set. This 'exactly one' rule is what separates a function from a general relation, and it lets functions be composed, inverted and graphed as curves.
Mental models at work here
- Connects 12 other ideas across 3 disciplines.
- A cross-disciplinary bridge — its connections reach into 5 other fields.
- Most of its connections are of the “Teaching link” kind.
- It exercises 1 reusable thinking pattern.
Derived from the graph’s real structure — observations, not a score.
Sources
- Wikipedia (English & German editions) verifiedmoderate evidence
- Wikidata verifiedmoderate evidence