Division Operation - Learning Module

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Use Cases for Division

Division in the Real World

The division operator might seem abstract, but it addresses a query pattern that appears constantly in real-world database applications. Every time a business asks "who has done all of X?" or "what satisfies every requirement in Y?", they're expressing a division query.

In this page, we'll explore concrete, industry-relevant use cases across multiple domains. You'll see that division isn't an academic curiosity—it's a fundamental operation that solves critical business problems. More importantly, understanding these patterns will help you recognize division opportunities in your own work, even when the query isn't immediately obvious.

What You Will Learn

By the end of this page, you'll be able to identify division patterns across diverse domains, understand how to model 'for all' requirements as division queries, and recognize the structural elements that make a problem suitable for the division operator.

E-Commerce and Retail

E-commerce platforms frequently need to identify customers, products, or transactions that satisfy complete sets of conditions. Let's explore several division patterns in this domain.

Pattern Recognition

In each use case, identify: (1) The dividend relation (contains the pairings), (2) The divisor relation (contains requirements), and (3) The result (entities satisfying all requirements).

Use Case 1: Complete Product Category Purchase

•Business Question: Which customers have purchased every product in the 'Basics' collection?
•Dividend: Purchases(CustomerID, ProductID) — all customer-product purchase records
•Divisor: BasicsCollection(ProductID) — products in the Basics collection
•Division: Purchases ÷ BasicsCollection → CustomerIDs who bought ALL basics products
•Business Value: Target these 'completionist' customers for upselling premium collections

Complete Collection PurchaseFind customers who purchased all products in the 'Starter Kit' collection.

Input

Purchases(CustomerID, ProductID):
| CustomerID | ProductID |
|------------|------------|
| C001       | P10        |
| C001       | P20        |
| C001       | P30        |
| C002       | P10        |
| C002       | P30        |
| C003       | P10        |
| C003       | P20        |
| C003       | P30        |
| C003       | P40        |

StarterKit(ProductID):
| ProductID |
|-----------|
| P10       |
| P20       |
| P30       |

Output

Purchases ÷ StarterKit:
| CustomerID |
|------------|
| C001       |
| C003       |

Analysis:
• C001: Has P10 ✓, P20 ✓, P30 ✓ → Included
• C002: Has P10 ✓, P20 ✗, P30 ✓ → Excluded (missing P20)
• C003: Has P10 ✓, P20 ✓, P30 ✓ (plus P40) → Included

Use Case 2: Warehouse Inventory Completeness

•Business Question: Which warehouses stock every essential item?
•Dividend: WarehouseInventory(WarehouseID, ItemID) — items stocked at each warehouse
•Divisor: EssentialItems(ItemID) — items classified as essential
•Division: WarehouseInventory ÷ EssentialItems → Warehouses with complete essential stock
•Business Value: Identify warehouses that can handle emergency orders independently

Use Case 3: Vendor Qualification

•Business Question: Which vendors can supply every component for Product X?
•Dividend: VendorSupplies(VendorID, ComponentID) — what each vendor can supply
•Divisor: ProductXComponents(ComponentID) — components needed for Product X
•Division: VendorSupplies ÷ ProductXComponents → Single-source vendors for Product X
•Business Value: Simplify supply chain by identifying vendors who can fulfill complete orders

Academic and Educational Systems

Educational institutions abound with "complete all requirements" queries. Graduation requirements, course prerequisites, and certification programs all naturally express as division operations.

The Classic Division Problem

Perhaps the most intuitive division use case: "Which students have completed all required courses for graduation?"

Schema:

Completed(StudentID, CourseID) — courses each student has passed
RequiredCourses(CourseID) — courses required for the degree

Query: Completed ÷ RequiredCourses

Result: StudentIDs who have passed EVERY required course

Extensions:

Track degree-specific requirements: partition RequiredCourses by major
Track credit hours: add constraints beyond simple completion
Track grades: filter Completed to include only passing grades first

Graduation Eligibility CheckDetermine which CS students have completed all core requirements.

Input

Completed(StudentID, CourseID):
| StudentID | CourseID |
|-----------|----------|
| S101      | CS101    |
| S101      | CS201    |
| S101      | CS301    |
| S101      | MATH101  |
| S102      | CS101    |
| S102      | CS201    |
| S103      | CS101    |
| S103      | CS201    |
| S103      | CS301    |
| S103      | CS401    |

CSCoreReqs(CourseID):
| CourseID |
|----------|
| CS101    |
| CS201    |
| CS301    |

Output

Completed ÷ CSCoreReqs:
| StudentID |
|-----------|
| S101      |
| S103      |

• S101: Has CS101 ✓, CS201 ✓, CS301 ✓ → Eligible
• S102: Has CS101 ✓, CS201 ✓, CS301 ✗ → Not eligible
• S103: Has CS101 ✓, CS201 ✓, CS301 ✓ → Eligible

Supply Chain and Manufacturing

Supply chain management is rich with division patterns. Bill-of-materials queries, supplier qualification, and inventory management all involve "has everything needed" logic.

Key Supply Chain Division Patterns

•Single-Source Suppliers: Which suppliers can provide ALL parts for a product?
•Complete Kits: Which warehouses have ALL components to assemble product X?
•Qualified Manufacturers: Which factories have ALL certifications for production?
•Full Capability: Which service centers can perform ALL required maintenance tasks?
•Regional Coverage: Which distributors serve ALL regions in a territory?

Bill of Materials (BOM) QueryFind suppliers who can provide all components for the 'Engine Assembly'.

Input

SupplierParts(SupplierID, PartID):
| SupplierID | PartID  |
|------------|----------|
| SUP01      | PISTON   |
| SUP01      | CRANKSHAFT|
| SUP01      | CAMSHAFT |
| SUP01      | VALVE    |
| SUP02      | PISTON   |
| SUP02      | CRANKSHAFT|
| SUP03      | PISTON   |
| SUP03      | CRANKSHAFT|
| SUP03      | CAMSHAFT |
| SUP03      | VALVE    |
| SUP03      | BEARING  |

EngineAssemblyBOM(PartID):
| PartID     |
|------------|
| PISTON     |
| CRANKSHAFT |
| CAMSHAFT   |
| VALVE      |

Output

SupplierParts ÷ EngineAssemblyBOM:
| SupplierID |
|------------|
| SUP01      |
| SUP03      |

Analysis:
• SUP01: Has all 4 required parts ✓
• SUP02: Missing CAMSHAFT and VALVE ✗
• SUP03: Has all 4 required parts (plus BEARING) ✓

Benefits of Single-Source

•Simplified procurement
•Single point of accountability
•Consolidated shipping
•Volume discounts
•Reduced coordination overhead

Risks of Single-Source

•Supply chain vulnerability
•Reduced negotiating power
•Quality concentration risk
•Capacity constraints
•Dependency on one relationship

Division for Risk Assessment

Run the division query with different part subsets to understand supply chain resilience. How many suppliers can provide 90% of parts? 80%? This extends beyond pure division but uses it as a foundation.

Security and Authorization

Access control and authorization systems frequently require "has all required permissions" checks. Division provides a clean model for these authorization queries.

Role-Based Access Control (RBAC)

•Schema: UserPermissions(UserID, PermissionID) — permissions granted to each user
•Schema: RequiredPermissions(PermissionID) — permissions needed for an action
•Query: UserPermissions ÷ RequiredPermissions
•Result: Users authorized to perform the action
•Pattern: This is the core of permission checking in many systems

Admin Panel Access ControlDetermine which users can access the Admin Panel (requires all admin permissions).

Input

UserPermissions(UserID, PermissionID):
| UserID | PermissionID    |
|--------|------------------|
| U001   | READ_USERS       |
| U001   | WRITE_USERS      |
| U001   | DELETE_USERS     |
| U001   | READ_LOGS        |
| U002   | READ_USERS       |
| U002   | WRITE_USERS      |
| U003   | READ_USERS       |
| U003   | WRITE_USERS      |
| U003   | DELETE_USERS     |
| U003   | READ_LOGS        |
| U003   | SYSTEM_CONFIG    |

AdminPanelPerms(PermissionID):
| PermissionID  |
|---------------|
| READ_USERS    |
| WRITE_USERS   |
| DELETE_USERS  |
| READ_LOGS     |

Output

UserPermissions ÷ AdminPanelPerms:
| UserID |
|--------|
| U001   |
| U003   |

• U001: Has all 4 required permissions ✓
• U002: Missing DELETE_USERS and READ_LOGS ✗
• U003: Has all 4 required permissions (plus SYSTEM_CONFIG) ✓

Advanced Authorization Patterns:

Multi-Factor Authorization:

RequiredFactors(FactorType): all factors needed (password, biometric, hardware key)
UserFactors(UserID, FactorType): factors a user has configured
UserFactors ÷ RequiredFactors → Users with complete MFA setup

Compliance Auditing:

RequiredTrainings(TrainingID): mandatory compliance trainings
CompletedTrainings(EmployeeID, TrainingID): completed trainings
CompletedTrainings ÷ RequiredTrainings → Compliant employees

Feature Flags with Prerequisites:

FeaturePrereqs(FeatureFlag): flags that must be enabled
UserFlags(UserID, FeatureFlag): flags enabled for user
UserFlags ÷ FeaturePrereqs → Users who can access new feature

Performance Consideration

In production authorization systems, division may be too expensive for real-time checks. The pattern is conceptually correct, but implementations often use cached permission sets or pre-computed authorization tables. Division is ideal for batch auditing and analysis.

Healthcare and Clinical Systems

Healthcare systems involve complex compliance and protocol requirements. Division naturally models "has completed all required steps" queries critical for patient safety and regulatory compliance.

Ensuring Complete Treatment

Scenario: A treatment protocol consists of multiple steps (consultations, tests, procedures). We need to identify patients who have completed ALL required steps.

Schema:

PatientProcedures(PatientID, ProcedureID) — procedures performed
ProtocolSteps(ProcedureID) — required steps in the protocol

Query: PatientProcedures ÷ ProtocolSteps

Result: Patients who have completed the entire protocol

Clinical Importance:

Treatment efficacy depends on complete protocols
Insurance reimbursement may require protocol completion
Quality metrics track protocol adherence

Project Management and Workflows

Project management involves tracking task completion, milestone achievement, and workflow progression—all areas where division excels.

Project Management Use Cases

•Milestone Completion: Which projects have completed all Phase 1 milestones?
•Team Qualification: Which teams have all required skill sets for Project X?
•Approval Workflows: Which documents have received all required approvals?
•Sprint Completion: Which sprints have all committed stories done?
•Release Readiness: Which releases have passed all quality gates?

Quality Gate VerificationFind releases that have passed all quality gates and are ready for production.

Input

ReleaseGates(ReleaseID, GateID):
| ReleaseID | GateID          |
|-----------|------------------|
| R2.1      | UNIT_TESTS       |
| R2.1      | INTEGRATION_TESTS|
| R2.1      | SECURITY_SCAN    |
| R2.1      | PERF_TEST        |
| R2.1      | STAGING_DEPLOY   |
| R2.2      | UNIT_TESTS       |
| R2.2      | INTEGRATION_TESTS|
| R2.2      | SECURITY_SCAN    |
| R2.3      | UNIT_TESTS       |
| R2.3      | INTEGRATION_TESTS|
| R2.3      | SECURITY_SCAN    |
| R2.3      | PERF_TEST        |
| R2.3      | STAGING_DEPLOY   |

ProductionGates(GateID):
| GateID           |
|------------------|
| UNIT_TESTS       |
| INTEGRATION_TESTS|
| SECURITY_SCAN    |
| PERF_TEST        |
| STAGING_DEPLOY   |

Output

ReleaseGates ÷ ProductionGates:
| ReleaseID |
|-----------|
| R2.1      |
| R2.3      |

• R2.1: Passed all 5 gates ✓
• R2.2: Missing PERF_TEST and STAGING_DEPLOY ✗
• R2.3: Passed all 5 gates ✓

Workflow State Management:

Document Approval Workflow:

Approvals(DocumentID, ApproverRole)
RequiredApprovals(ApproverRole)  // e.g., Legal, Finance, Executive
Approvals ÷ RequiredApprovals → Fully approved documents

Agile Story Completion:

StoryTasks(StoryID, TaskID)
DefinitionOfDone(TaskID)  // e.g., Code, Review, Test, Deploy
StoryTasks ÷ DefinitionOfDone → Truly complete stories

Onboarding Completion:

OnboardingProgress(EmployeeID, StepID)
OnboardingSteps(StepID)  // e.g., Paperwork, Training, Setup
OnboardingProgress ÷ OnboardingSteps → Fully onboarded employees

Recognizing Division Patterns

With these diverse examples, let's crystallize how to recognize when division is the appropriate solution.

Division Pattern Indicators

•'All' in the question: 'All required', 'every', 'complete set', 'full coverage'
•Two-level structure: Entities related to requirements through a pairing relation
•Containment test: Does entity's set contain all of requirement set?
•Universal quantification: For ALL x in requirements, condition holds
•NOT 'in at least one': That's existential (join), not universal (division)

Quick Pattern Recognition Guide
Question Phrase	Likely Operation	Example
'in at least one'	Join or Exists	Students in any CS course
'in some'	Join or Selection	Suppliers with some certification
'in all/every'	Division	Students in ALL required courses
'has the complete'	Division	Warehouses with complete inventory
'satisfies all'	Division	Candidates meeting all criteria
'none of'	Set Difference	Students not in any course

The Mental Model

When you see 'Which X have ALL of Y?', immediately think: X-Y pairings ÷ Y list. The pairing relation is your dividend, the requirement set is your divisor.

Summary: Division Use Cases

We've explored division across diverse domains. Here are the key patterns and insights:

Key Takeaways

•Division is ubiquitous — 'For all' patterns appear across e-commerce, education, healthcare, security, and more
•Common structure: Entity-Requirement pairings ÷ Requirements → Qualifying Entities
•Key phrases: 'all required', 'every', 'complete', 'full' signal division needs
•Not counting: Division is binary (pass/fail), not a coverage percentage
•Practical applications: Authorization, compliance, graduation, supply chain, QA workflows
•Pattern recognition is crucial: Identifying division saves complex SQL workarounds

What's Next:

Now that you can recognize division use cases, we'll dive into how division actually works. The next page covers the division algorithm—the step-by-step procedure to compute division using other relational operators.

Use Cases Mastered

You can now identify division patterns across diverse domains. You understand the structural elements—pairing relations and requirement sets—that make problems suitable for division. Next, we'll learn how to compute division step by step.

Use Cases for Division

Division in the Real World

What You Will Learn

E-Commerce and Retail

E-commerce platforms frequently need to identify customers, products, or transactions that satisfy complete sets of conditions. Let's explore several division patterns in this domain.

Pattern Recognition

In each use case, identify: (1) The dividend relation (contains the pairings), (2) The divisor relation (contains requirements), and (3) The result (entities satisfying all requirements).

Use Case 1: Complete Product Category Purchase

•Business Question: Which customers have purchased every product in the 'Basics' collection?
•Dividend: Purchases(CustomerID, ProductID) — all customer-product purchase records
•Divisor: BasicsCollection(ProductID) — products in the Basics collection
•Division: Purchases ÷ BasicsCollection → CustomerIDs who bought ALL basics products
•Business Value: Target these 'completionist' customers for upselling premium collections

Complete Collection PurchaseFind customers who purchased all products in the 'Starter Kit' collection.

Input

Purchases(CustomerID, ProductID):
| CustomerID | ProductID |
|------------|------------|
| C001       | P10        |
| C001       | P20        |
| C001       | P30        |
| C002       | P10        |
| C002       | P30        |
| C003       | P10        |
| C003       | P20        |
| C003       | P30        |
| C003       | P40        |

StarterKit(ProductID):
| ProductID |
|-----------|
| P10       |
| P20       |
| P30       |

Output

Purchases ÷ StarterKit:
| CustomerID |
|------------|
| C001       |
| C003       |

Analysis:
• C001: Has P10 ✓, P20 ✓, P30 ✓ → Included
• C002: Has P10 ✓, P20 ✗, P30 ✓ → Excluded (missing P20)
• C003: Has P10 ✓, P20 ✓, P30 ✓ (plus P40) → Included

Use Case 2: Warehouse Inventory Completeness

•Business Question: Which warehouses stock every essential item?
•Dividend: WarehouseInventory(WarehouseID, ItemID) — items stocked at each warehouse
•Divisor: EssentialItems(ItemID) — items classified as essential
•Division: WarehouseInventory ÷ EssentialItems → Warehouses with complete essential stock
•Business Value: Identify warehouses that can handle emergency orders independently

Use Case 3: Vendor Qualification

•Business Question: Which vendors can supply every component for Product X?
•Dividend: VendorSupplies(VendorID, ComponentID) — what each vendor can supply
•Divisor: ProductXComponents(ComponentID) — components needed for Product X
•Division: VendorSupplies ÷ ProductXComponents → Single-source vendors for Product X
•Business Value: Simplify supply chain by identifying vendors who can fulfill complete orders

Academic and Educational Systems

Educational institutions abound with "complete all requirements" queries. Graduation requirements, course prerequisites, and certification programs all naturally express as division operations.

The Classic Division Problem

Perhaps the most intuitive division use case: "Which students have completed all required courses for graduation?"

Schema:

Completed(StudentID, CourseID) — courses each student has passed
RequiredCourses(CourseID) — courses required for the degree

Query: Completed ÷ RequiredCourses

Result: StudentIDs who have passed EVERY required course

Extensions:

Track degree-specific requirements: partition RequiredCourses by major
Track credit hours: add constraints beyond simple completion
Track grades: filter Completed to include only passing grades first

Graduation Eligibility CheckDetermine which CS students have completed all core requirements.

Input

Completed(StudentID, CourseID):
| StudentID | CourseID |
|-----------|----------|
| S101      | CS101    |
| S101      | CS201    |
| S101      | CS301    |
| S101      | MATH101  |
| S102      | CS101    |
| S102      | CS201    |
| S103      | CS101    |
| S103      | CS201    |
| S103      | CS301    |
| S103      | CS401    |

CSCoreReqs(CourseID):
| CourseID |
|----------|
| CS101    |
| CS201    |
| CS301    |

Output

Completed ÷ CSCoreReqs:
| StudentID |
|-----------|
| S101      |
| S103      |

• S101: Has CS101 ✓, CS201 ✓, CS301 ✓ → Eligible
• S102: Has CS101 ✓, CS201 ✓, CS301 ✗ → Not eligible
• S103: Has CS101 ✓, CS201 ✓, CS301 ✓ → Eligible

Supply Chain and Manufacturing

Supply chain management is rich with division patterns. Bill-of-materials queries, supplier qualification, and inventory management all involve "has everything needed" logic.

Key Supply Chain Division Patterns

•Single-Source Suppliers: Which suppliers can provide ALL parts for a product?
•Complete Kits: Which warehouses have ALL components to assemble product X?
•Qualified Manufacturers: Which factories have ALL certifications for production?
•Full Capability: Which service centers can perform ALL required maintenance tasks?
•Regional Coverage: Which distributors serve ALL regions in a territory?

Bill of Materials (BOM) QueryFind suppliers who can provide all components for the 'Engine Assembly'.

Input

SupplierParts(SupplierID, PartID):
| SupplierID | PartID  |
|------------|----------|
| SUP01      | PISTON   |
| SUP01      | CRANKSHAFT|
| SUP01      | CAMSHAFT |
| SUP01      | VALVE    |
| SUP02      | PISTON   |
| SUP02      | CRANKSHAFT|
| SUP03      | PISTON   |
| SUP03      | CRANKSHAFT|
| SUP03      | CAMSHAFT |
| SUP03      | VALVE    |
| SUP03      | BEARING  |

EngineAssemblyBOM(PartID):
| PartID     |
|------------|
| PISTON     |
| CRANKSHAFT |
| CAMSHAFT   |
| VALVE      |

Output

SupplierParts ÷ EngineAssemblyBOM:
| SupplierID |
|------------|
| SUP01      |
| SUP03      |

Analysis:
• SUP01: Has all 4 required parts ✓
• SUP02: Missing CAMSHAFT and VALVE ✗
• SUP03: Has all 4 required parts (plus BEARING) ✓

Benefits of Single-Source

•Simplified procurement
•Single point of accountability
•Consolidated shipping
•Volume discounts
•Reduced coordination overhead

Risks of Single-Source

•Supply chain vulnerability
•Reduced negotiating power
•Quality concentration risk
•Capacity constraints
•Dependency on one relationship

Division for Risk Assessment

Security and Authorization

Access control and authorization systems frequently require "has all required permissions" checks. Division provides a clean model for these authorization queries.

Role-Based Access Control (RBAC)

•Schema: UserPermissions(UserID, PermissionID) — permissions granted to each user
•Schema: RequiredPermissions(PermissionID) — permissions needed for an action
•Query: UserPermissions ÷ RequiredPermissions
•Result: Users authorized to perform the action
•Pattern: This is the core of permission checking in many systems

Admin Panel Access ControlDetermine which users can access the Admin Panel (requires all admin permissions).

Input

UserPermissions(UserID, PermissionID):
| UserID | PermissionID    |
|--------|------------------|
| U001   | READ_USERS       |
| U001   | WRITE_USERS      |
| U001   | DELETE_USERS     |
| U001   | READ_LOGS        |
| U002   | READ_USERS       |
| U002   | WRITE_USERS      |
| U003   | READ_USERS       |
| U003   | WRITE_USERS      |
| U003   | DELETE_USERS     |
| U003   | READ_LOGS        |
| U003   | SYSTEM_CONFIG    |

AdminPanelPerms(PermissionID):
| PermissionID  |
|---------------|
| READ_USERS    |
| WRITE_USERS   |
| DELETE_USERS  |
| READ_LOGS     |

Output

UserPermissions ÷ AdminPanelPerms:
| UserID |
|--------|
| U001   |
| U003   |

• U001: Has all 4 required permissions ✓
• U002: Missing DELETE_USERS and READ_LOGS ✗
• U003: Has all 4 required permissions (plus SYSTEM_CONFIG) ✓

Advanced Authorization Patterns:

Multi-Factor Authorization:

RequiredFactors(FactorType): all factors needed (password, biometric, hardware key)
UserFactors(UserID, FactorType): factors a user has configured
UserFactors ÷ RequiredFactors → Users with complete MFA setup

Compliance Auditing:

RequiredTrainings(TrainingID): mandatory compliance trainings
CompletedTrainings(EmployeeID, TrainingID): completed trainings
CompletedTrainings ÷ RequiredTrainings → Compliant employees

Feature Flags with Prerequisites:

FeaturePrereqs(FeatureFlag): flags that must be enabled
UserFlags(UserID, FeatureFlag): flags enabled for user
UserFlags ÷ FeaturePrereqs → Users who can access new feature

Performance Consideration

Healthcare and Clinical Systems

Healthcare systems involve complex compliance and protocol requirements. Division naturally models "has completed all required steps" queries critical for patient safety and regulatory compliance.

Ensuring Complete Treatment

Scenario: A treatment protocol consists of multiple steps (consultations, tests, procedures). We need to identify patients who have completed ALL required steps.

Schema:

PatientProcedures(PatientID, ProcedureID) — procedures performed
ProtocolSteps(ProcedureID) — required steps in the protocol

Query: PatientProcedures ÷ ProtocolSteps

Result: Patients who have completed the entire protocol

Clinical Importance:

Treatment efficacy depends on complete protocols
Insurance reimbursement may require protocol completion
Quality metrics track protocol adherence

Project Management and Workflows

Project management involves tracking task completion, milestone achievement, and workflow progression—all areas where division excels.

Project Management Use Cases

•Milestone Completion: Which projects have completed all Phase 1 milestones?
•Team Qualification: Which teams have all required skill sets for Project X?
•Approval Workflows: Which documents have received all required approvals?
•Sprint Completion: Which sprints have all committed stories done?
•Release Readiness: Which releases have passed all quality gates?

Quality Gate VerificationFind releases that have passed all quality gates and are ready for production.

Input

ReleaseGates(ReleaseID, GateID):
| ReleaseID | GateID          |
|-----------|------------------|
| R2.1      | UNIT_TESTS       |
| R2.1      | INTEGRATION_TESTS|
| R2.1      | SECURITY_SCAN    |
| R2.1      | PERF_TEST        |
| R2.1      | STAGING_DEPLOY   |
| R2.2      | UNIT_TESTS       |
| R2.2      | INTEGRATION_TESTS|
| R2.2      | SECURITY_SCAN    |
| R2.3      | UNIT_TESTS       |
| R2.3      | INTEGRATION_TESTS|
| R2.3      | SECURITY_SCAN    |
| R2.3      | PERF_TEST        |
| R2.3      | STAGING_DEPLOY   |

ProductionGates(GateID):
| GateID           |
|------------------|
| UNIT_TESTS       |
| INTEGRATION_TESTS|
| SECURITY_SCAN    |
| PERF_TEST        |
| STAGING_DEPLOY   |

Output

ReleaseGates ÷ ProductionGates:
| ReleaseID |
|-----------|
| R2.1      |
| R2.3      |

• R2.1: Passed all 5 gates ✓
• R2.2: Missing PERF_TEST and STAGING_DEPLOY ✗
• R2.3: Passed all 5 gates ✓

Workflow State Management:

Document Approval Workflow:

Approvals(DocumentID, ApproverRole)
RequiredApprovals(ApproverRole)  // e.g., Legal, Finance, Executive
Approvals ÷ RequiredApprovals → Fully approved documents

Agile Story Completion:

StoryTasks(StoryID, TaskID)
DefinitionOfDone(TaskID)  // e.g., Code, Review, Test, Deploy
StoryTasks ÷ DefinitionOfDone → Truly complete stories

Onboarding Completion:

OnboardingProgress(EmployeeID, StepID)
OnboardingSteps(StepID)  // e.g., Paperwork, Training, Setup
OnboardingProgress ÷ OnboardingSteps → Fully onboarded employees

Recognizing Division Patterns

With these diverse examples, let's crystallize how to recognize when division is the appropriate solution.

Division Pattern Indicators

•'All' in the question: 'All required', 'every', 'complete set', 'full coverage'
•Two-level structure: Entities related to requirements through a pairing relation
•Containment test: Does entity's set contain all of requirement set?
•Universal quantification: For ALL x in requirements, condition holds
•NOT 'in at least one': That's existential (join), not universal (division)

Quick Pattern Recognition Guide
Question Phrase	Likely Operation	Example
'in at least one'	Join or Exists	Students in any CS course
'in some'	Join or Selection	Suppliers with some certification
'in all/every'	Division	Students in ALL required courses
'has the complete'	Division	Warehouses with complete inventory
'satisfies all'	Division	Candidates meeting all criteria
'none of'	Set Difference	Students not in any course

The Mental Model

When you see 'Which X have ALL of Y?', immediately think: X-Y pairings ÷ Y list. The pairing relation is your dividend, the requirement set is your divisor.

Summary: Division Use Cases

We've explored division across diverse domains. Here are the key patterns and insights:

Key Takeaways

•Division is ubiquitous — 'For all' patterns appear across e-commerce, education, healthcare, security, and more
•Common structure: Entity-Requirement pairings ÷ Requirements → Qualifying Entities
•Key phrases: 'all required', 'every', 'complete', 'full' signal division needs
•Not counting: Division is binary (pass/fail), not a coverage percentage
•Practical applications: Authorization, compliance, graduation, supply chain, QA workflows
•Pattern recognition is crucial: Identifying division saves complex SQL workarounds

What's Next:

Use Cases Mastered