Set Operations in Relational Algebra

Context: FIT2094_MOC · union, intersect, difference on relations · require union-compatible operands · the FIT1058 set operations applied to data

Quick Revision

• 🎯 Objective: set ops on relation bodies ➔ union $\cup$, intersect $\cap$, difference $-$.
• 📦 Core Components: require union-compatible operands (same arity, compatible domains).
• ⚡ Key Constraint: difference is not symmetric ($A-B\ne B-A$); duplicates auto-collapse.

📝 Core

1. The Operations

• Union ➔ $A\cup B$ = tuples in either (duplicates once).
• Intersection ➔ $A\cap B$ = tuples in both.
• Difference ➔ $A-B$ = tuples in $A$ not $B$.

2. Union-Compatibility

• Same arity ➔ same number of attributes.
• Compatible domains ➔ positionally matching domains.
• Mandatory ➔ else the result isn’t a well-formed relation.

3. Symmetry

• Symmetric ➔ union, intersection commute.
• Asymmetric ➔ difference: $A-B\ne B-A$.

Key identities:

$\text{STOREA}=\{1,2,3\},\text{STOREB}=\{1,2,33\}(\text{by product\_id})$ $A\cup B=\{1,2,3,33\},A\cap B=\{1,2\},A-B=\{3\},B-A=\{33\}$

⚖️ Core Decision Matrix

Operation	Symbol	Symmetric?	Result
union	$\cup$	yes	either
intersection	$\cap$	yes	both
difference	$-$	no	in A not B
precondition	—	—	union-compatible

When It Flips: these are exactly the FIT1058 set operations on the tuple sets — inclusion–exclusion and De Morgan carry over, with union-compatibility as the new precondition. Closure holds, so set ops compose with σ/π and joins. Unlike joins (different schemas), set ops need identical schemas.

📊 Exam Execution Trace

Manual Execution Trace

STOREA$=\{1,2,3\}$, STOREB$=\{1,2,33\}$:

Step / State	Op	Result
0 (Init)	—	—
1	$A\cup B$	$\{1,2,3,33\}$
2	$A\cap B$	$\{1,2\}$
3	$A-B$	$\{3\}$

⚠️ Common Mistakes

• 💡 Union-compatibility required ➔ same arity + positionally compatible domains (both (product_id, product_name)); difference is directional.

🧠 Active Recall

Define union, intersection, and difference on relations, and the union-compatibility requirement.

• Hint: Set ops + compatibility.

Answer

• Short answer: $\cup$ = either, $\cap$ = both, $-$ = in A not B; operands must be union-compatible (same arity, compatible domains).
• Why: Well-formed ➔ compatibility ensures the result is a valid relation.

Which set operation is not symmetric, and how do these relate to FIT1058 set theory?

• Hint: Difference directional.

Answer

• Short answer: Difference: $A-B\ne B-A$; union/intersection commute.
• Why: Tuple sets ➔ these are FIT1058 set operations on relation bodies plus union-compatibility.