Symmetric function

Key takeaways

Symmetric function 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 function to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Symmetric function from memory before moving on to harder problems.

Reference excerpt

In mathematics, a function of n {\displaystyle n} variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f ( x 1 , x 2 ) {\displaystyle f\left(x_{1},x_{2}\right)} of two arguments is a symmetric function if and only if f ( x 1 , x 2 ) = f ( x 2 , x 1 ) {\displaystyle f\left(x_{1},x_{2}\right)=f\left(x_{2},x_{1}\right)} for all x 1 {\displaystyle x_{1}} and x 2 {\displaystyle x_{2}} such that ( x 1 , x 2 ) {\displaystyle \left(x_{1},x_{2}\right)} and ( x 2 , x 1 ) {\displaystyle \left(x_{2},x_{1}\right)} are in the domain of f . {\displaystyle f.} The most commonly encountered symmetric functions are polynomial functions, which are given by the symmetric polynomials. A related notion is alternating polynomials, which change sign under an interchange of variables. Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric k {\displaystyle k} -tensors on a vector space V {\displaystyle V} is isomorphic to the space of homogeneous polynomials of degree k {\displaystyle k} on V . {\displaystyle V.} Symmetric functions should not be confused with even and odd functions, which have a different sort of symmetry.

Symmetrization

Given any function f {\displaystyle f} in n {\displaystyle n} variables with values in an abelian group, a symmetric function can be constructed by summing values of f {\displaystyle f} over all permutations of the arguments. Similarly, an anti-symmetric function can be constructed by summing over even permutations and subtracting the sum over odd permutations. These operations are of course not invertible, and could well result in a function that is identically zero for nontrivial functions f . {\displaystyle f.} The only general case where f {\displaystyle f} can be recovered if both its symmetrization and antisymmetrization are known is when n = 2 {\displaystyle n=2} and the abelian group admits a division by 2 (inverse of doubling); then f {\displaystyle f} is equal to half the sum of its symmetrization and its antisymmetrization.

Examples

Consider the real function

f ( x 1 , x 2 , x 3 ) = ( x − x 1 ) ( x − x 2 ) ( x − x 3 ) . {\displaystyle f(x_{1},x_{2},x_{3})=(x-x_{1})(x-x_{2})(x-x_{3}).}

By definition, a symmetric function with n {\displaystyle n} variables has the property that

f ( x 1 , x 2 , … , x n ) = f ( x 2 , x 1 , … , x n ) = f ( x 3 , x 1 , … , x n , x n − 1 ) , etc. {\displaystyle f(x_{1},x_{2},\ldots ,x_{n})=f(x_{2},x_{1},\ldots ,x_{n})=f(x_{3},x_{1},\ldots ,x_{n},x_{n-1}),\quad {\text{ etc.}}}

In general, the function remains the same for every permutation of its variables. This means that, in this case,

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Symmetric function

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

In research: Symmetric function 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 function 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 function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Combinatorics, Properties of binary operations, Symmetric functions, so understanding it makes those chapters shorter.
In everyday life: Look for Symmetric function 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 Symmetric function in 20 minutes

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

Frequently asked questions

What is Symmetric function in simple terms?

In mathematics, a function of n {\displaystyle n} variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f ( x 1 , x 2 ) {\displaystyle f\left(x_{1},x_{2}\right)} of two arguments is a symmetric function if and only if f ( x 1 , x 2 ) = f ( x 2…

Why does Symmetric function 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 function?

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 function.