Problem: Which of the following binary operations are closed?
- subtraction of positive integers
- division of nonzero integers
- function composition of polynomials with real coefficient
- multiplication of $2 \times 2$ matrices with real entires
- exponentiation of integers
Solution: Let $S$ be a set and $*$ be a binary operation on $S$. The binary operation $*$ is said to be closed if for all $a, b \in S$, the element $a*b$ is also in $S$.
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Subtraction of positive integers is not closed. For example, \( 1 - 2 = -1 \) which is not a positive integer.
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Division of nonzero integers is not closed. For example, \( 1 \div 2 = \frac{1}{2} \) which is not an integer.
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Function composition of polynomials with real coefficients is closed. Let us prove this. Let
\[
p(x) = \sum_{i=0}^{n} a_i x^i \quad \text{and} \quad q(x) = \sum_{j=0}^{m} b_j x^j.
\]
Then
\begin{align*}
p(q(x)) & = p\left(\sum_{j=0}^{m} b_j x^j\right) \\
& = \sum_{i=0}^{n} a_i \left(\sum_{j=0}^{m} b_j x^j\right)^i \\
& = \sum_{i=0}^{n} a_i \left(\sum_{j=0}^{m} b_j x^j\right) \left(\sum_{j=0}^{m} b_j x^j\right) \cdots \left(\sum_{j=0}^{m} b_j x^j\right) \quad \text{(i times)} \\
& = \sum_{i=0}^{n} a_i \sum_{j_1=0}^{m} \sum_{j_2=0}^{m} \cdots \sum_{j_i=0}^{m} b_{j_1} b_{j_2} \cdots b_{j_i} x^{j_1 + j_2 + \cdots + j_i}.
\end{align*}
This is a polynomial with real coefficients, so function composition of polynomials with real coefficients is closed.
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Multiplication of \( 2 \times 2 \) matrices with real entries is closed. Let
\[
A =
\begin{bmatrix}
a & b \\
c & d \\
\end{bmatrix} \text{ and } B =
\begin{bmatrix}
x & y \\
z & w \\
\end{bmatrix}.
\]
Then,
\begin{align*}
AB & =
\begin{bmatrix}
a & b \\
c & d \\
\end{bmatrix}
\begin{bmatrix}
x & y \\
z & w \\
\end{bmatrix} \\
& =
\begin{bmatrix}
ax + bz & ay + bw \\
cx + dz & cy + dw \\
\end{bmatrix}.
\end{align*}
Thus, the product of two \( 2 \times 2 \) matrices with real entries is a \( 2 \times 2 \) matrix with real entries.
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Exponentiation of integers is not closed. For example, \( 2^{-1} = \frac{1}{2} \) which is not an integer.