Reflexive closure

Key takeaways

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

Reference excerpt

In mathematics, the reflexive closure of a binary relation R {\displaystyle R} on a set X {\displaystyle X} is the smallest reflexive relation on X {\displaystyle X} that contains R {\displaystyle R} , i.e. the set R ∪ { ( x , x ) ∣ x ∈ X } {\displaystyle R\cup \{(x,x)\mid x\in X\}} . For example, if X {\displaystyle X} is a set of distinct numbers and x R y {\displaystyle xRy} means " x {\displaystyle x} is less than y {\displaystyle y} ", then the reflexive closure of R {\displaystyle R} is the relation " x {\displaystyle x} is less than or equal to y {\displaystyle y} ".

Definition The reflexive closure S {\displaystyle S} of a relation R {\displaystyle R} on a set X {\displaystyle X} is given by

S = R ∪ { ( x , x ) ∣ x ∈ X } {\displaystyle S=R\cup \{(x,x)\mid x\in X\}}

In plain English, the reflexive closure of R {\displaystyle R} is the union of R {\displaystyle R} with the identity relation on X . {\displaystyle X.}

Example As an example, if

X = { 1 , 2 , 3 , 4 } {\displaystyle X=\{1,2,3,4\}}

R = { ( 1 , 1 ) , ( 1 , 3 ) , ( 2 , 2 ) , ( 3 , 3 ) , ( 4 , 4 ) } {\displaystyle R=\{(1,1),(1,3),(2,2),(3,3),(4,4)\}}

then the relation R {\displaystyle R} is already reflexive by itself, so it does not differ from its reflexive closure. However, if any of the reflexive pairs in R {\displaystyle R} was absent, it would be inserted for the reflexive closure. For example, if on the same set X {\displaystyle X}

R = { ( 1 , 1 ) , ( 1 , 3 ) , ( 2 , 2 ) , ( 4 , 4 ) } {\displaystyle R=\{(1,1),(1,3),(2,2),(4,4)\}}

then the reflexive closure is

S = R ∪ { ( x , x ) ∣ x ∈ X } = { ( 1 , 1 ) , ( 1 , 3 ) , ( 2 , 2 ) , ( 3 , 3 ) , ( 4 , 4 ) } . {\displaystyle S=R\cup \{(x,x)\mid x\in X\}=\{(1,1),(1,3),(2,2),(3,3),(4,4)\}.}

See also Symmetric closure Transitive closure – Smallest transitive relation containing a given binary relation

References

Franz Baader and Tobias Nipkow, Term Rewriting and All That, Cambridge University Press, 1998, p. 8

Worked examples

Example 1 — a first encounter with Reflexive closure

Start with the simplest possible case. Write down what Reflexive closure claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Reflexive closure 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 Reflexive closure 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 Reflexive closure

In research: Reflexive closure appears in science 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 Reflexive closure 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: Reflexive closure is common in secondary-school and first-year university syllabi. It links to neighbouring topics Binary relations, Closure operators, Programming language theory stubs, so understanding it makes those chapters shorter.
In everyday life: Look for Reflexive closure 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.

How to study Reflexive closure in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Reflexive closure 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 Reflexive closure out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Reflexive closure in simple terms?

In mathematics, the reflexive closure of a binary relation R {\displaystyle R} on a set X {\displaystyle X} is the smallest reflexive relation on X {\displaystyle X} that contains R {\displaystyle R} , i.e. the set R ∪ { ( x , x ) ∣ x ∈ X } {\displaystyle R\cup \{(x,x)\mid x\in X\}} . For example…

Why does Reflexive closure matter?

Because it connects several science 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 Reflexive closure?

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 Reflexive closure.