Question

Determine which operations polynomials, integers, and rationals are not closed under.a. Polynomials:b. Integers:c. Rationals:

Answer 1

We say that a set A is closed under a certain operation if

$\begin{gathered} f,g\in A \\ \Rightarrow f\oplus g\in A \end{gathered}$

In our case,

a) It is evident that the polynomials are closed under addition and subtraction

$\begin{gathered} (a_mx^m+a_{m-1}x^{m-1}+\ldots+a_1x+a_0)+(b_nx^n+b_{n-1}x^{n-1}+\ldots+b_1x+b_0)_{} \\ =a_mx^m+\cdots(a_n+b_n)x_n+\cdots(a_0+b_0) \end{gathered}$

Which is a polynomial. Similarly, in the case of the multiplication of polynomials.

However, in the case of the division of polynomials, the result is not always a polynomial.

$\begin{gathered} \frac{x^2+3x+2}{x-1}\to\text{not a polynomial} \\ \frac{x^2+3x+2}{x+1}=x+3\to polynomial \end{gathered}$

Thus, the set of polynomials is not closed under division.

b) Notice that the integers are closed under addition and subtraction (the integers are ...-2,-1,0,1,2,...).

$\begin{gathered} a,b\in\text{integers} \\ \Rightarrow a+b\to\text{integer} \\ \end{gathered}$

Similarly, in the case of the multiplication of integers.

Nevertheless, in the case of the division of integers,

$\begin{gathered} \frac{6}{3}=2\to\text{integer} \\ \frac{7}{2}=3.5\to\text{not an integer} \end{gathered}$

Therefore, the set of integers is not closed under division.

c) Rational numbers are numbers of the form a/b, where a and b are integers.

$\begin{gathered} \frac{a}{b}+\frac{c}{d}=\frac{ad+bc\to\text{integer}}{bd\to integer}\to\text{rational number} \\ \frac{a}{b}\cdot\frac{c}{d}=\frac{ac\to integer}{bd\to\text{integer}}\to\text{rational number} \end{gathered}$

Once again, the set of rational numbers is closed under addition, subtraction and multiplication. As for the division of rational numbers,

$\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{ad\to integer}{bc\to integer}\to rational\text{ number}$

The rational numbers are also closed under division.

There are no restrictions for the set of rational numbers regarding operations.