mathematics

Symmetric set

Symmetric set 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 Symmetric set rather than just read about it. In short: In mathematics, a nonempty subset S of a group G is said to be symmetric if it contains the inverses of all of its elements. Definition In set notation a subset S {\displaystyle S} of a group G {\displaystyle G} is called symmetric if whenever s ∈ S {\displaystyle s\in S} then the inverse of s {\displaystyle s} also belongs to S . {\displaystyle S.} So if G {\displaystyle G} is written multiplicatively then S {\disp…

Key takeaways

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

Reference excerpt

In mathematics, a nonempty subset S of a group G is said to be symmetric if it contains the inverses of all of its elements.

Definition In set notation a subset S {\displaystyle S} of a group G {\displaystyle G} is called symmetric if whenever s ∈ S {\displaystyle s\in S} then the inverse of s {\displaystyle s} also belongs to S . {\displaystyle S.} So if G {\displaystyle G} is written multiplicatively then S {\displaystyle S} is symmetric if and only if S = S − 1 {\displaystyle S=S^{-1}} where S − 1 := { s − 1 : s ∈ S } . {\displaystyle S^{-1}:=\left\{s^{-1}:s\in S\right\}.} If G {\displaystyle G} is written additively then S {\displaystyle S} is symmetric if and only if S = − S {\displaystyle S=-S} where − S := { − s : s ∈ S } . {\displaystyle -S:=\{-s:s\in S\}.}

If S {\displaystyle S} is a subset of a vector space then S {\displaystyle S} is said to be a symmetric set if it is symmetric with respect to the additive group structure of the vector space; that is, if S = − S , {\displaystyle S=-S,} which happens if and only if − S ⊆ S . {\displaystyle -S\subseteq S.} The symmetric hull of a subset S {\displaystyle S} is the smallest symmetric set containing S , {\displaystyle S,} and it is equal to S ∪ − S . {\displaystyle S\cup -S.} The largest symmetric set contained in S {\displaystyle S} is S ∩ − S . {\displaystyle S\cap -S.}

Sufficient conditions Arbitrary unions and intersections of symmetric sets are symmetric. Any vector subspace in a vector space is a symmetric set.

Examples In R , {\displaystyle \mathbb {R} ,} examples of symmetric sets are intervals of the type ( − k , k ) {\displaystyle (-k,k)} with k > 0 , {\displaystyle k>0,} and the sets Z {\displaystyle \mathbb {Z} } and ( − 1 , 1 ) . {\displaystyle (-1,1).}

If S {\displaystyle S} is any subset of a group, then S ∪ S − 1 {\displaystyle S\cup S^{-1}} and S ∩ S − 1 {\displaystyle S\cap S^{-1}} are symmetric sets. Any balanced subset of a real or complex vector space is symmetric.

See also Absolutely convex set – Convex and balanced set Absorbing set – Set that can be "inflated" to reach any point Balanced function – Construct in functional analysisPages displaying short descriptions of redirect targets Balanced set – Construct in functional analysis Bounded set (topological vector space) – Generalization of boundedness Convex set – In geometry, set whose intersection with every line is a single line segment Minkowski functional – Function made from a set Star domain – Property of point sets in Euclidean spaces

References

This article incorporates material from symmetric set on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Worked examples

Example 1 — a first encounter with Symmetric set

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

In research: Symmetric set 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 Symmetric set 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 set is common in secondary-school and first-year university syllabi. It links to neighbouring topics Group theory, Linear algebra, so understanding it makes those chapters shorter.
In everyday life: Look for Symmetric set 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 Symmetric set in 20 minutes

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

Frequently asked questions

What is Symmetric set in simple terms?

In mathematics, a nonempty subset S of a group G is said to be symmetric if it contains the inverses of all of its elements. Definition In set notation a subset S {\displaystyle S} of a group G {\displaystyle G} is called symmetric if whenever s ∈ S {\displaystyle s\in S} then the inverse of s {\di…

Why does Symmetric set 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 Symmetric set?

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 set.