science

Symmetric relation

Symmetric relation is a science 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 Symmetric relation rather than just read about it. In short: A symmetric relation is a type of binary relation. A homogeneous relation R {\displaystyle R} on a set X {\displaystyle X} is symmetric if: for all a {\displaystyle a} and b {\displaystyle b} in X {\displaystyle X} , a R b {\displaystyle aRb} if and only if b R a {\displaystyle bRa} , where the notation a R b {\displaystyle aRb} means that the ordered pair ( a , b ) {\displaystyle (a,b)} is in the relation R {\displ…

Key takeaways

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

Reference excerpt

A symmetric relation is a type of binary relation. A homogeneous relation R {\displaystyle R} on a set X {\displaystyle X} is symmetric if:

for all a {\displaystyle a} and b {\displaystyle b} in X {\displaystyle X} , a R b {\displaystyle aRb} if and only if b R a {\displaystyle bRa} , where the notation a R b {\displaystyle aRb} means that the ordered pair ( a , b ) {\displaystyle (a,b)} is in the relation R {\displaystyle R} . An example is the relation "is equal to", because if a = b is true then b = a is also true. If RT represents the converse of R, then R is symmetric if and only if R = RT. Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.

Examples

In mathematics "is equal to" (equality) (whereas "is less than" is not symmetric) "is comparable to", for elements of a partially ordered set "... and ... are odd":

Outside mathematics "is married to" (in most legal systems) "is a fully biological sibling of" "has coauthored a book with" "is a homophone of" "is a co-worker of" "is a teammate of"

Relationship to asymmetric and antisymmetric relations

By definition, a nonempty relation cannot be both symmetric and asymmetric (where if a is related to b, then b cannot be related to a (in the same way)). However, a relation can be neither symmetric nor asymmetric, which is the case for "is less than or equal to" and "preys on"). Symmetric and antisymmetric (where the only way a can be related to b and b be related to a is if a = b) are actually independent of each other, as these examples show.

Properties A symmetric and transitive relation is always quasireflexive. One way to count the symmetric relations on n elements, that in their binary matrix representation the upper right triangle determines the relation fully, and it can be arbitrary given, thus there are as many symmetric relations as n × n binary upper triangle matrices, 2n(n+1)/2.

Note that S(n, k) refers to Stirling numbers of the second kind.

Notes

References

See also Commutative property – Property of some mathematical operations Symmetry in mathematics Symmetry – Mathematical invariance under transformations

Illustrations

Worked examples

Example 1 — a first encounter with Symmetric relation

Start with the simplest possible case. Write down what Symmetric relation 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 Symmetric 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 Symmetric 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 Symmetric relation

In research: Symmetric relation 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 Symmetric 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: Symmetric relation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Properties of binary relations, Symmetric relations, so understanding it makes those chapters shorter.
In everyday life: Look for Symmetric 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.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Symmetric relation in 20 minutes

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

Frequently asked questions

What is Symmetric relation in simple terms?

A symmetric relation is a type of binary relation. A homogeneous relation R {\displaystyle R} on a set X {\displaystyle X} is symmetric if: for all a {\displaystyle a} and b {\displaystyle b} in X {\displaystyle X} , a R b {\displaystyle aRb} if and only if b R a {\displaystyle bRa} , where the not…

Why does Symmetric relation 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 Symmetric 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 Symmetric relation.