notebook/notes/combinatorics/additive-principle.md

---
title: Additive Principle
TARGET DECK: Obsidian::STEM
FILE TAGS: combinatorics set
tags:
  - combinatorics
  - set
---

## Overview

The **additive principle** states that two finite and disjoint sets $A$ and $B$ satisfy $$|A \cup B| = |A| + |B|$$

This can be generalized to any number of finite and disjoint sets in the obvious way.

%%ANKI
Basic
What does the additive principle state?
Back: Given finite and disjoint sets $A$ and $B$, $|A \cup B| = |A| + |B|$.
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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END%%

%%ANKI
Basic
The additive property applies to sets exhibiting what two properties?
Back: Finiteness and disjointedness.
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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END%%

%%ANKI
Basic
Why does $|A \cup B| \neq |A| + |B|$ in the general sense?
Back: Members of $A \cap B$ are counted twice erroneously.
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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END%%

%%ANKI
Basic
Which C construct corresponds to the additive property?
Back: `union`
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
Tags: c17
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END%%

%%ANKI
Basic
How do we denote $A$ and $B$ are disjoint using standard set notation?
Back: $A \cap B = \varnothing$
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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END%%

## Bibliography

* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).