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} (x^2+3x+2)/(x-1)\to\text{not a polynomial} \\ (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} (6)/(3)=2\to\text{integer} \\ (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} (a)/(b)+(c)/(d)=\frac{ad+bc\to\text{integer}}{bd\to integer}\to\text{rational number} \\ (a)/(b)\cdot(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,

`((a)/(b))/((c)/(d))=(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.