Problem 68 - Discrete Mathematics with Applications

Question

For Exercises \(68-73,\) use the following definition of a simple algebraic expression: $$\langle\text {expression}\rangle : :=\langle\text { term }\rangle |\langle\text { sign }\rangle\langle\text { term }\rangle |$$ $$\langle\text { expression }\rangle\langle\text { adding operator }\rangle\langle\text { term }\rangle$$ $$\langle\operatorname{sign}\rangle \therefore=+ 1-$$ $$\langle\text { adding operator}\rangle: :=+1-$$ $$\langle\text { term }\rangle : :=\langle\text { factor }\rangle |$$ $$\langle\text { term }\rangle\langle\text { multiplying operator }\rangle\langle\text { factor }\rangle$$ $$\langle\text { multiplying operator }\rangle := *| /$$ $$\langle\text { factor }\rangle : :=\langle\text { letter }|\rangle (\langle\text { expression }\rangle |\langle\text { expression }\rangle$$ $$\langle\text { letter }\rangle : := a|b| c | \ldots : z$$ Determine if each is a legal expression. $$a+b *(c / d)$$

Step-by-Step Solution

Answer

The given expression \(a + b*(c / d)\) is a legal algebraic expression as it satisfies all the rules provided in the problem statement. It contains legal terms, factors, and adding and multiplying operators.

Step 1: Expression

The given algebraic expression is: \(a + b*(c / d)\)

Step 1: Identify Terms

Identify the terms in the given expression: - Term 1: \(a\) - Term 2: \(b*(c / d)\)

Step 2: Check Adding Operators

Determine if the adding operators are legal according to the defined rules: Adding Operator 1: \(+\), which is legal according to the given rules.

Step 3: Identify Factors within Terms

Identify the factors within each term: - Term 1 Factor: \(a\) (letter) - Term 2 Factor 1: \(b\) (letter) - Term 2 Factor 2: \(c / d\) (expression)

Step 4: Check Multiplying Operators

Check if the multiplying operators within the terms are legal according to the defined rules: Multiplying Operator 1: \(*\), which is legal according to the given rules.

Step 5: Check Expression within Factors

Check if the expression in the factors are legal: Factor Expression: \(c / d\) This expression contains one term with two factors as \(c\) and \(d\), and a legal multiplying operator \( /\) according to the given rules. Therefore, the expression is legal.

Step 7: Conclusion

After analyzing the given expression \(a + b*(c / d)\), we can conclude that it satisfies all the rules provided in the problem statement and is a legal algebraic expression.

Key Concepts

Syntax of Expressions

Understanding the syntax of algebraic expressions is crucial in mathematics. Syntax refers to the set of rules that define the structure of an expression. In our scenario, a "simple algebraic expression" is systematically defined, allowing us to understand what components are valid and how they must be arranged.

The core structure includes terms and operators linked in specific orders. A complete expression could be a straightforward term or a combination of terms and operators that comply with the syntax rules.

This structured nature ensures clarity and consistency across mathematical problems, making it possible to communicate and solve algebraic problems effectively.

Mathematical Operators

Mathematical operators play a key role in forming expressions and conducting arithmetic operations. They are symbols used to perform specific operations on one or more operands in an expression.

In simpler terms, operators are the action parts of mathematical expressions. Commonly used operators fall into two main categories:

• Adding Operators: These include the plus "+" and minus "-" signs used to add or subtract values.
• Multiplying Operators: These include the asterisk "*" and slash "/", representing multiplication and division respectively.

In the given expression, these operators link terms like glue, following predefined rules to ensure the expression is valid.

Legal Expressions

A legal expression is an algebraic expression that adheres to defined rules and structural syntax. For an expression to be considered legal, every component must fit according to the prescribed patterns and permissions, such as those for operators and terms.

The expression in question, which is given as \(a + b*(c / d)\), is legal because it complies with the rule:

• It uses legal adding and multiplying operators (i.e., "+" and "*").
• Each term and factor is aligned as per the given syntax rules.

Understanding and identifying legal expressions help in ensuring accuracy and effectiveness in solving mathematical equations and problems.

Terms and Factors

In algebra, terms and factors are fundamental components that form expressions. A term could be a single factor or multiple factors connected by multiplying operators. Being clear about what is a term and what is a factor helps unravel the structure of algebraic expressions.

In our example, the expression \(a + b*(c / d)\) consists of:

• Term 1: \(a\), a simple letter factor.
• Term 2: \(b*(c / d)\), which itself includes multipliable factors.

In this setup, "b" is a straightforward factor, whereas "c/d" is a more complex factor encapsulated within parentheses.

Understanding these distinctions allows for easier manipulation and evaluation of algebraic expressions, paving the way for solving complex equations.