Relation (Database)

Context: FIT2094_MOC · the central object of the Relational Model · heading (schema) + body (tuples) · sized by degree and cardinality

Quick Revision

• 🎯 Objective: a structured set of data ➔ heading (schema) + body (tuples).
• 📦 Core Components: heading $R(A_1,\dots ,A_n)$ ➔ body $\{t_1,\dots ,t_m\}$.
• ⚡ Key Constraint: degree ($n$ attributes, fixed) vs cardinality ($m$ tuples, varies).

📝 Core

1. Two Parts

• Heading (schema) ➔ $R(A_1,\dots ,A_n)$ — name + attributes, each from a domain.
• Body (instance) ➔ $r(R)=\{t_1,\dots ,t_m\}$ — the current set of tuples, changing over time.

2. Tuple, Degree, Cardinality

• Tuple ➔ ordered list $t=⟨v_1,\dots ,v_n⟩$.
• Degree ➔ $n$ = number of attributes (heading property).
• Cardinality ➔ $m$ = number of tuples (body property, fluctuates).

3. Writing a Relation

• Singular name ➔ it denotes a set.
• PK underlined ➔ $\text{STAFF}(\underline{\text{staff\_id}},\dots )$.

Key identities:

$\text{CUSTOMER}(\underline{\text{custno}},\text{custname},\text{custadd},\text{custcredlimit})(\text{heading, degree }4)$ $r(\text{CUSTOMER})=\{t_1,t_2,t_3\}(\text{body, cardinality }3)$

⚖️ Core Decision Matrix

Term	Meaning	Belongs to
heading	schema $R(A_1,\dots ,A_n)$	structure
body	tuples $\{t_1,\dots ,t_m\}$	data
degree	# attributes $n$	heading
cardinality	# tuples $m$	body

When It Flips: because the body is a set, no duplicate tuples and tuples/attributes are unordered (Relation Properties). An $n$-degree relation is a subset of the Cartesian Product of its attribute domains — the n-ary Relation.

📊 Exam Execution Trace

Manual Execution Trace

CUSTOMER with 3 rows:

Step / State	Quantity	Value
0 (Init)	—	—
1	degree	4 (attributes)
2	cardinality	3 (tuples)
3	PK	custno (underlined)

⚠️ Common Mistakes

• 💡 Degree fixed, cardinality varies ➔ degree is a design-time schema property; cardinality is run-time data that changes as rows are inserted/deleted.

🧠 Active Recall

Define the heading and body of a relation, and the terms degree and cardinality.

• Hint: Schema vs instance.

Answer

• Short answer: Heading = $R(A_1,\dots ,A_n)$ (structure); body = $\{t_1,\dots ,t_m\}$ (data); degree = $n$ attributes; cardinality = $m$ tuples.
• Why: Fixed vs varying ➔ degree from the heading, cardinality from the body.

How is a relation written, and how does it correspond to the n-ary relation?

• Hint: Singular + underlined PK.

Answer

• Short answer: Singular name, PK underlined; a degree-$n$ relation is a subset of the product of $n$ domains.
• Why: Same terms ➔ degree (attributes) and cardinality (tuples) match the n-ary Relation.