# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/11/mathematics/function-composition-inverse-function/function-and-non-function
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/function-composition-inverse-function/function-and-non-function/en.mdx

Output docs content for large language models.

---

import { Diagram, RelationVisualizer } from "./diagram";

export const metadata = {
  title: "Function and Non-Function",
  description: "Distinguish functions from non-functions using arrow diagrams and mathematical definitions. Learn one-to-one mapping rules with visual examples.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/27/2025",
  subject: "Function Composition and Inverse Function",
};

## Relations Between Sets

In mathematics, a **relation** from a set <InlineMath math="A" /> to a set <InlineMath math="B" /> is a rule that connects members of set <InlineMath math="A" /> with members of set <InlineMath math="B" />. This pairing can be in any form.

**Example:**

The "less than" relation between <InlineMath math="A = \{1, 2, 3\}" /> and <InlineMath math="B = \{1, 2, 3, 4\}" /> yields the pairs <InlineMath math="(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)" />.

**Explanation:**

We look for all pairs <InlineMath math="(a, b)" /> with <InlineMath math="a \in A" /> and <InlineMath math="b \in B" /> where <InlineMath math="a < b" />.

- For <InlineMath math="a=1" /> -> <InlineMath math="1 < 2" />, <InlineMath math="1 < 3" />, <InlineMath math="1 < 4" />. Pairs: <InlineMath math="(1, 2), (1, 3), (1, 4)" />.
- For <InlineMath math="a=2" /> -> <InlineMath math="2 < 3" />, <InlineMath math="2 < 4" />. Pairs: <InlineMath math="(2, 3), (2, 4)" />.
- For <InlineMath math="a=3" /> -> <InlineMath math="3 < 4" />. Pair: <InlineMath math="(3, 4)" />.

The combination of all these pairs is <InlineMath math="\{(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)\}" />.

## Functions as Special Relations

A **function** (or mapping) <InlineMath math="f" /> from a set <InlineMath math="A" /> to a set <InlineMath math="B" />, written <InlineMath math="f: A \to B" />, is a special relation that satisfies two conditions:

1.  **Every** element <InlineMath math="x \in A" /> must have a pair <InlineMath math="y \in B" />.

    <BlockMath math="\forall x \in A, \exists y \in B \text{ such that } (x, y) \in f" />

2.  Every element <InlineMath math="x \in A" /> has **exactly one** pair <InlineMath math="y \in B" />.

    <BlockMath math="\text{If } (x, y_1) \in f \text{ and } (x, y_2) \in f, \text{ then } y_1 = y_2" />

This means every member of the domain must be connected, and cannot have more than one connection.

## Arrow Diagram Examples

Here are visual examples of relations using arrow diagrams to distinguish between functions and non-functions.

### Relations That Are Not Functions

<div className="flex flex-col gap-4">
  <Diagram
    title="One to Many"
    description="Element b has more than one pair (m and n)."
  >
    <RelationVisualizer
      domain={[
        { id: "a", label: "a" },
        { id: "b", label: "b" },
        { id: "c", label: "c" },
      ]}
      codomain={[
        { id: "m", label: "m" },
        { id: "n", label: "n" },
      ]}
      mappings={[
        { from: "a", to: "m" },
        { from: "b", to: "m" },
        { from: "b", to: "n" },
        { from: "c", to: "n" },
      ]}
    />
  </Diagram>

  <Diagram
    title="Domain Element Without a Pair"
    description="Element c does not have a pair in the codomain."
  >
    <RelationVisualizer
      domain={[
        { id: "a", label: "a" },
        { id: "b", label: "b" },
        { id: "c", label: "c" },
      ]}
      codomain={[
        { id: "m", label: "m" },
        { id: "n", label: "n" },
        { id: "o", label: "o" },
      ]}
      mappings={[
        { from: "a", to: "m" },
        { from: "b", to: "o" },
        // 'c' has no mapping
      ]}
    />
  </Diagram>
</div>

### Relations That Are Functions

<Diagram
  title="Exactly One Pair"
  description="Each domain element (p, q, r) has exactly one pair."
>
  <RelationVisualizer
    domain={[
      { id: "p", label: "p" },
      { id: "q", label: "q" },
      { id: "r", label: "r" },
    ]}
    codomain={[
      { id: "x", label: "x" },
      { id: "y", label: "y" },
      { id: "z", label: "z" },
    ]}
    mappings={[
      { from: "p", to: "x" },
      { from: "q", to: "y" },
      { from: "r", to: "y" }, // Many-to-one is allowed for functions
    ]}
  />
</Diagram>