Problem 18 - Discrete Mathematics with Applications

Question

Using Example 11.1 , determine if each is a well-formed and fully parenthesized arithmetic expression. \((x \uparrow((y-x) \uparrow(-z)))\)

Step-by-Step Solution

Answer

The given expression \((x \uparrow((y \sim x) \uparrow(-z)))\) is well-formed and fully parenthesized, as all three pairs of parentheses form a valid structure and all operands are properly enclosed within these pairs.

Step 1: Identify parentheses pairs

First, let's look at the expression and identify all the opening and closing parentheses: $$(x \uparrow((y \sim x) \uparrow(-z)))$$ The expression contains three opening parentheses '(' and three closing parentheses ')' which is a good sign. Now, we need to verify if they form valid pairs.

Step 2: Verify parentheses pairing

We'll now check if the parentheses pairs create a valid nesting for the expression: $$(x \uparrow(\underline{(}(y \sim x)\underline{)} \uparrow(-z)))$$ Looks like the first pair of parentheses form a valid nested structure. Let's continue to the subsequent pairs: $$(x \uparrow((y \sim x) \uparrow\underline{(}(-z)\underline{)}))$$ The second pair is also well-formed. The last pair is: $$\underline{(}(x \uparrow((y \sim x) \uparrow(-z)))\underline{)}$$ All three pairs of parentheses form a valid structure for the expression.

Step 3: Ensure operands are properly enclosed

Finally, let's confirm that the operands within each pair of parentheses are properly enclosed. We can break down the expression as follows: 1. \((x \uparrow(\cdots))\): The operand \(x\) is enclosed by the outer pair of parentheses, and it is related to the internal expression with the operator \(\uparrow\). This is well-formed. 2. \(((y \sim x) \uparrow(\cdots))\): The expression \((y \sim x)\) is enclosed by a pair of parentheses, and it is related to the internal expression with the operator \(\uparrow\). This is well-formed as well. 3. \((y \sim x)\): The operands \(y\) and \(x\) are enclosed by a pair of parentheses and related to the operator \(\sim\). This is well-formed. 4. \((-z)\): The operand \(-z\) is enclosed by a pair of parentheses. This is well-formed.

Step 4: Conclusion

After analyzing the expression, we can conclude that it is a well-formed and fully parenthesized arithmetic expression. All parentheses pairs create a valid structure, and all operands are properly enclosed within these pairs.

Key Concepts

Parentheses Pairing

In the context of arithmetic expressions, parentheses play a crucial role in determining the order of operations and ensuring clarity. To verify an expression is correctly formed, we first need to focus on the concept of parentheses pairing. This means checking whether each opening parenthesis has a corresponding closing parenthesis and if they create a valid nested structure.

• Opening and closing parentheses must be equal in number.
• Every opening parenthesis should find its matching closing partner in a grammatically correct sequence.
• Nesting should be proper, meaning that a closing parenthesis should match the most recent unmatched opening parenthesis.

In our example expression, \[x \uparrow((y \sim x) \uparrow(-z))\]we have a perfect match of three opening and three closing parentheses. Each set is properly nested and balanced, indicating a well-formed structure.

Well-formed Expressions

A well-formed expression is an expression that adheres to certain syntactic rules without any ambiguity. In arithmetic expressions, this means:

• Parentheses must enclose operands and operators properly, thereby specifying clear precedence.
• Operators require an appropriate number of operands.
• No unmatched parentheses or incomplete operations.

In our examined expression,\[x \uparrow((y \sim x) \uparrow(-z))\]every component contributes to making the expression well-formed. For instance:- The outer parenthesis fully encase the expression, indicating independent evaluation.- The sub-expression \((y \sim x)\) is neatly encapsulated, ensuring clarity in operation precedence.- Similarly, \((-z)\) is properly enclosed.Such attention to detail avoids confusion and ensures the expression is computationally valid.

Arithmetic Expressions

An arithmetic expression involves numbers, variables, and operators like addition, subtraction, multiplication, and division. For more complex operations like exponentiation and negation, parentheses are used to dictate the order of these operations. Additionally, symbols like\[\uparrow\]and \[\sim\]in the given expression indicate operations such as exponentiation and negation.Key features of arithmetic expressions include:

• Combining operators and operands to perform calculations.
• Utilizing parentheses for modifying operation sequences.
• Transforming logically correct subsections into comprehensive, computationally effective sequences.

The expression \(x \uparrow((y \sim x) \uparrow(-z))\) is solved by resolving the innermost parentheses first, then moving outward, respecting the hierarchy indicated by the parentheses.