Sets
Declare what belongs to a collection and the universe within which membership is judged.
SRAI Book 1 · Chapter 2 · PU-B01-C02
Sets, Logic, Relations and Functions
Sets determine what belongs. Logic determines truth conditions. Relations determine what is connected. Functions determine how declared inputs map to outputs. Lesson 2 connects these foundations to data filters, databases, knowledge systems, algorithms and transparent decision rules.
01 / LEARNING OUTCOMES
Make categories, rules and mappings explicit.
- Distinguish membership from subsethood.
- Compute and interpret core set operations.
- Evaluate logical connectives and material implication.
- Distinguish relations, partial functions and total functions.
- Translate predicates into reproducible filters.
- Document universes, domains, codomains, missing-data rules and decision reasons.
02 / FOUNDATIONAL STRUCTURE
Four ideas organise much of data and computation.
Logic
State truth conditions and make decision rules inspectable.
Relations
Represent declared connections between elements without assuming a unique output.
Functions
Map each declared input to exactly one output in a stated codomain.
03 / SET OPERATIONS
Membership rules become reproducible operations.
For sets A and B, the notebook verifies union, intersection, difference, complement relative to a declared universe, and Cartesian product.
The universe is part of the specification. Without it, a complement is ambiguous and the resulting analytical claim cannot be defended.
CORE NOTATION
| Union | A ∪ B |
| Intersection | A ∩ B |
| Difference | A ∖ B |
| Cartesian product | A × B |
Every operation is checked against explicit expected results.
A connection may be one-to-many, many-to-one or incomplete.
A relation records which ordered pairs belong. It does not automatically guarantee a unique output for every input.
Every declared input has exactly one output.
Totality and uniqueness are obligations that must be tested—not merely assumed from a label or data structure.
04 / DEFENSIBLE DECISION RULES
A transparent rule records both the outcome and why it follows.
- Declare the universe and eligibility population.
- State predicates and truth conditions explicitly.
- Define treatment of missing and exceptional values.
- Separate a relation from a function claim.
- Verify totality and uniqueness where required.
- Preserve reasons so decisions can be reviewed.
CONTROLLED RESOURCES
Read, reproduce, practise and review.
READ
View the complete GitHub production unit ↗
Audited Chapter 2
Controlled publication edition.
PDF ↗ REPRODUCECanonical notebook
Executable and independently verified.
IPYNB ↗ PRACTISEExercises and solutions
Progress toward independent application.
PDF ↗ RUNOpen in Google Colab
Execute the approved notebook in a clean cloud runtime.
COLAB ↗ DOWNLOADComplete release package
Chapter, notebook, exercises, Executive Brief, code, metadata and validation records.
ZIP ↓ APPLYExecutive brief
Connect analytical controls to decisions.
PDF ↗