mathematics

Nullary relation

Nullary relation is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Nullary relation rather than just read about it. In short: In mathematics, a nullary relation, 0-ary relation, or relation of degree zero is a relation with zero attributes. There are exactly two relations of degree zero.

Key takeaways

Nullary relation belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Nullary relation to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Nullary relation from memory before moving on to harder problems.

Reference excerpt

In mathematics, a nullary relation, 0-ary relation, or relation of degree zero is a relation with zero attributes. There are exactly two relations of degree zero. One has cardinality zero; that is, contains no tuples at all. The other has cardinality 1 and contains only the unique 0-tuple.:56 The zero-degree relations represent true and false in relational algebra.:57 Under the closed-world assumption, an n-ary relation is interpreted as the extension of some n-adic predicate: all and only those n-tuples whose values, substituted for corresponding free variables in the predicate, yield propositions that hold true, appear in the relation. A zero-degree relation is therefore interpreted as the extension of the 0-adic predicate P() → true. The zero-degree relation with cardinality zero therefore represents false because it contains no tuples that yield a true proposition, and the zero-degree relation with cardinality 1 represents true because it contains the unique 0-tuple that yields a true proposition. The zero-degree relations are also significant as identities for certain operators in the relational algebra. The zero-degree relation of cardinality 1 is the identity with respect to join (⋈); that is, when it is joined with any other relation R, the result is R. Defining an identity with respect to join makes it possible to extend the binary join operator into a n-ary join operator.:89 Since the relational Cartesian product is a special case of join, the zero-degree relation of cardinality 1 is also the identity with respect to the Cartesian product.:89 A projection of a relation over no attributes yields one of the relations of degree zero. If the projected relation has cardinality 0, the projection will have cardinality 0; if the projected relation has positive cardinality, the result will have cardinality 1. Hugh Darwen refers to the zero-degree relation with cardinality 0 as TABLE_DUM and the relation with cardinality 1 as TABLE_DEE, alluding to the characters Tweedledum and Tweedledee.

See also Empty set Identity element Relational algebra

References

Worked examples

Example 1 — a first encounter with Nullary relation

Start with the simplest possible case. Write down what Nullary relation claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Nullary relation before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Nullary relation ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Nullary relation

In research: Nullary relation appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Nullary relation in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Nullary relation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Relational algebra, so understanding it makes those chapters shorter.
In everyday life: Look for Nullary relation outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Nullary relation in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Nullary relation means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Nullary relation out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Nullary relation in simple terms?

In mathematics, a nullary relation, 0-ary relation, or relation of degree zero is a relation with zero attributes. There are exactly two relations of degree zero.

Why does Nullary relation matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Nullary relation?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Nullary relation.