itemset concept

Definition	A set of items that appear together in a transaction. A closed itemset has no superset with the same support and a maximal one has no frequent superset. Chapter 5.
Example	Bread and butter together is an itemset, and its support is the number of baskets holding both.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `itemset`, kind concept.
Status	no ledger row names this entry. Corrections: none recorded.
Defining equation	Book equation 5.1.
Assumptions and scope	A set of items that appear together in a transaction. Its support lies in the unit interval, the empty itemset has support one, a superset has support at most that of a subset, and the support of a union is at most the smaller of the two supports. A closed itemset has no strict superset with the same support and a maximal frequent itemset has no frequent strict superset, and every maximal frequent itemset is closed, since a superset with the same support would be frequent too.
Prior art	none recorded
Evidence	`lean/DataMiningAsObservation/ItemSet.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

A set of items, with the transactions that contain it as its support.

Equation

Book equation 5.1.

\[s(X)=\frac{\sigma(X)}{N},\qquad c(X\to Y)=\frac{\sigma(X\cup Y)}{\sigma(X)}=\frac{s(X\cup Y)}{s(X)}.\]

Book equation 5.3.

\[X\subseteq Y\ \Longrightarrow\ s(X)\ge s(Y).\]

Conditions

A set of items that appear together in a transaction. Its support lies in the unit interval, the empty itemset has support one, a superset has support at most that of a subset, and the support of a union is at most the smaller of the two supports.
A closed itemset has no strict superset with the same support and a maximal frequent itemset has no frequent strict superset, and every maximal frequent itemset is closed, since a superset with the same support would be frequent too.

Conditions are curated in entries.toml rather than read from a record.

Ledger

none

First stated

Agrawal, Imieliński, and Swami, mining association rules, 1993, as TSK chapter 5 presents it and chapter 5 section 5.1 of Data Mining as Observation states it.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 5 section 5.1 to 5.3, 5.5	support, confidence, lift, the Apriori principle, FP-growth, closed and maximal itemsets, objective measures, Simpson’s paradox, cross-support	TSK 2e chapter 5

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/ItemSet.lean, theorems supportFrac_mem_unit, supportFrac_empty, supportFrac_anti, supportFrac_union_le, closed_of_maximal, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation chapters 5.

support; Apriori principle; safe pruning; confidence.

Status

Generated 2026-09-10 by encyclopedia/generate.py; book at observation-data-mining f3914f0; the commit of every record is listed in the encyclopedia’s provenance.

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