Answer
By starting with real numbers (base expressions) and repeatedly applying the recursive rules for addition, subtraction, multiplication, and division, we have explicit derivations showing:
1. \(\bigl((2 \cdot (0.3 - 4.2)) + (-7)\bigr)\) is in the set of arithmetic expressions.
2. \(\bigl((9 \cdot (6.1 + 2)) / ((4 - 7) \cdot 6)\bigr)\) is in the set of arithmetic expressions.
Hence both expressions satisfy the grammar for arithmetic expressions over the real numbers.
Work Step by Step
Below are **sample derivations** showing that each given expression is indeed a valid arithmetic expression under the stated grammar rules:
---
## Grammar Rules Recap
1. **Base Rule**:
Any real number \(r\) (positive, negative, or zero) is an arithmetic expression.
2. **Recursive Rules**: If \(u\) and \(v\) are arithmetic expressions, then so are:
- \((u + v)\)
- \((u - v)\)
- \((u \cdot v)\) (often written \((u * v)\))
- \(\bigl(u / v\bigr)\)
3. **Restriction**:
No other forms are allowed except those obtained from (1) and (2).
Below, “\(\cdot\)” will denote multiplication for clarity.
---
## (a) \(\bigl(\,(2 \cdot (0.3 - 4.2)) + (-7)\bigr)\)
We want to show \(\bigl(\,(2 \cdot (0.3 - 4.2)) + (-7)\bigr)\) is an arithmetic expression.
1. By the **base rule**, the real numbers \(2\), \(0.3\), \(4.2\), and \(-7\) are each arithmetic expressions.
2. Since \(0.3\) and \(4.2\) are expressions, apply the subtraction rule \((u - v)\) to get:
\[
(0.3 - 4.2),
\]
which is an arithmetic expression.
3. Since \(2\) is an expression and \((0.3 - 4.2)\) is an expression, apply the multiplication rule \((u \cdot v)\) to get:
\[
(2 \cdot (0.3 - 4.2)),
\]
which is an arithmetic expression.
4. Finally, \((2 \cdot (0.3 - 4.2))\) is an expression, and \((-7)\) (a real number) is an expression, so apply the addition rule \((u + v)\) to get:
\[
\bigl((2 \cdot (0.3 - 4.2)) + (-7)\bigr).
\]
This shows \(\bigl((2 \cdot (0.3 - 4.2)) + (-7)\bigr)\) is an arithmetic expression.
---
## (b) \(\bigl((9 \cdot (6.1 + 2)) \;/\; ((4 - 7) \cdot 6)\bigr)\)
We want to show \(\bigl((9 \cdot (6.1 + 2)) / ((4 - 7) \cdot 6)\bigr)\) is an arithmetic expression.
1. By the **base rule**, the real numbers \(9\), \(6.1\), \(2\), \(4\), \(7\), and \(6\) are arithmetic expressions.
2. From \(6.1\) and \(2\) being expressions, apply the addition rule \((u + v)\) to get:
\[
(6.1 + 2),
\]
an arithmetic expression.
3. From \(9\) and \((6.1 + 2)\) both being expressions, apply the multiplication rule \((u \cdot v)\) to get:
\[
(9 \cdot (6.1 + 2)).
\]
That is an arithmetic expression.
4. From \(4\) and \(7\) being expressions, apply the subtraction rule \((u - v)\) to get:
\[
(4 - 7),
\]
which is an arithmetic expression.
5. From \((4 - 7)\) and \(6\) both being expressions, apply the multiplication rule to get:
\[
((4 - 7) \cdot 6).
\]
That is an arithmetic expression.
6. Finally, we have \(\bigl(9 \cdot (6.1 + 2)\bigr)\) and \(\bigl((4 - 7) \cdot 6\bigr)\) as expressions, so apply the division rule \(\bigl(u / v\bigr)\) to form:
\[
\bigl((9 \cdot (6.1 + 2)) \;/\; ((4 - 7) \cdot 6)\bigr).
\]
This shows \(\bigl((9 \cdot (6.1 + 2)) / ((4 - 7) \cdot 6)\bigr)\) is an arithmetic expression.
