Question

What are some examples of a symmetric function?

Answer 1

To solve this question we need to know the concept of symmetric function. Symmetric function is a function having several variables which remain unchanged for any type of permutation of the variable. We will check whether the function is the same or not after interchanging the independent values of the function.

Complete step-by-step answer:
The question asks us to give some examples for symmetric functions. A function of a certain number of variables is systematic if the value remains the same no matter whatever is the order of the argument. To give a good knowledge about the symmetric function we can have a look at the given function. Consider the function $f$ which is the function of $x$ and $y$. Mathematically it is presented as $f\left( x,y \right)$ . If the function given is the same as $f\left( y,x \right)$. So for a function to be symmetric the condition is:
$f\left( x,y \right)=f\left( y,x \right)$
Some of the examples of symmetric function are:
(i) Consider a function $f\left( x,y \right)={{x}^{2}}+xy+{{y}^{2}}-{{r}^{2}}$ . If $x$ and $y$ are interchanged the function we get is $f\left( y,x \right)={{y}^{2}}+yx+{{x}^{2}}-{{r}^{2}}$ , Now we can see that both the function $f\left( x,y \right)$ and $f\left( y,x \right)$ are equal or same.
(ii) Consider a function $f\left( x,y,z \right)=\left( p-x \right)\left( p-y \right)\left( p-z \right)$ . If $x$, $y$ and $z$ are interchanged the function we get is $f\left( y,z,x \right)=\left( p-y \right)\left( p-z \right)\left( p-x \right)$ and $f\left( z,x,y \right)=\left( p-z \right)\left( p-x \right)\left( p-y \right)$, Now we can see that all the three function $f\left( x,y,z \right)$, $f\left( y,z,x \right)$ and $f\left( z,x,y \right)$ are equal.
(iii) Consider a function $f\left( x,y,z \right)={{x}^{2}}+{{y}^{2}}+{{z}^{2}}+\dfrac{1}{xyz}$ . If $x$, $y$ and $z$ are interchanged the function we get is $f\left( y,z,x \right)={{y}^{2}}+{{z}^{2}}+{{x}^{2}}+\dfrac{1}{yzx}$ and $f\left( z,x,y \right)={{z}^{2}}+{{x}^{2}}+{{y}^{2}}+\dfrac{1}{zxy}$, Now we can see that all the three function $f\left( x,y,z \right)$, $f\left( y,z,x \right)$ and $f\left( z,x,y \right)$ are equal.
$\therefore$ Above are the three examples of symmetric functions.

Note: Always remember that a symmetric function should not change under any permutation of its independent variables. To clarify the point let us take an instance, if a function has $3$ independent variables then each of the variables when interchanged should give the same value. All the three functions formed by interchanging the variable should be equal. If any of the functions is not equal then the function is not a symmetric function.