Problem 73 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The derivation tree for the given expression \(\mathbf{a} *(\mathbf{b}+\mathbf{c} / \mathbf{d})\) can be constructed as follows: ``` expression | term | ______________|_____________ | | factor multiplying operator | | a * | term | expression | ______________|_____________ | | term adding operator | | factor + | | b term | factor | | / | factor | c ``` This tree helps to visualize the structure of the expression, with each level representing a composition.

Step 1: 1. Start with the given expression

We have: $$\mathbf{a} *(\mathbf{b}+\mathbf{c} / \mathbf{d})$$

Step 2: 2. Break down the expression into terms

Following the rules for the expression: - First term: \(\mathbf{a}\) - Second term: \((\mathbf{b}+\mathbf{c} / \mathbf{d})\)

Step 3: 3. Identify the adding operator between terms

In this case, there is no adding operator between the two terms; there is only the multiplying operator \(*\).

Step 4: 4. Identify the multiplying operator between factors

According to the rules: - First term has no multiplying operator: \(\mathbf{a}\) - Second term has a multiplying operator: \(\mathbf{b}+\mathbf{c} / \mathbf{d}\)

Step 5: 5. Break down the second term into sub-expressions

Now, we can further break down the second term according to the rules: - First sub-expression: \(\mathbf{b}\) - Second sub-expression: \(\mathbf{c} / \mathbf{d}\)

Step 6: 6. Identify adding operators in sub-expressions

In this case, the adding operator for the second term \(+\) connects sub-expressions \(\mathbf{b}\) and \(\mathbf{c} / \mathbf{d}\).

Step 7: 7. Construct the derivation tree

Now, we construct the derivation tree: ``` expression | term | ______________|_____________ | | factor multiplying operator | | a * | term | expression | ______________|_____________ | | term adding operator | | factor + | | b term | factor | | / | factor | c ``` In this tree, each level represents a composition in the expression.

Key Concepts

Understanding Simple Algebraic Expressions

A simple algebraic expression consists of terms, which can be factors or combinations of factors connected by operators. To fully understand these expressions, it's crucial to recognize the structure based on specific grammar rules.

For instance, in the given expression \(\mathbf{a} *(\mathbf{b}+\mathbf{c} / \mathbf{d})\), "a" is a factor while "b+c/d" is a more complex term. Each term may be connected with an adding or multiplying operator.

Understanding this setup makes it easier to build and analyze the expression, as well as to construct a derivation tree that visually represents the structure.

Grammar Rules in Algebraic Expressions

Grammar rules guide how components of an expression fit together. In formal language theory, these rules define how terms and operators are sequenced. These rules ensure expressions are both valid and solvable.

In this exercise, the grammar is defined as:

• **Expression:** Consisting of terms, which are either standalone factors or composed with operators.
• **Adding operator:** Either "+" or "-"
• **Multiplying operator:** "*" or "/"

By following these grammar rules, you can confidently build expressions and derive their structures.

The Multiplying Operator in Expressions

In simple algebraic expressions, the multiplying operator \( *\) is used to connect factors or terms. This operator dictates how factors interact and combine.

For example, in the expression \(\mathbf{a} *(\mathbf{b}+\mathbf{c} / \mathbf{d})\), \(\mathbf{a}\) is multiplied by the entire term \((\mathbf{b}+\mathbf{c} / \mathbf{d})\).

When constructing a derivation tree, the multiplying operator acts as a branch point connecting these factors under a single term. This visualization highlights how elements within the expression relate through multiplication.

Exploring the Adding Operator

An adding operator in algebraic expressions includes both "\(+\)" and "\(-\)" symbols. It's used to connect sub-expressions or terms.

Within a term, like \((\mathbf{b}+\mathbf{c} / \mathbf{d})\), the adding operator \(+\) connects \(\mathbf{b}\) and the sub-expression \(\mathbf{c} / \mathbf{d}\).

In a derivation tree, this operator is a critical junction point, illustrating how distinct sub-expressions integrate into a unified term. By understanding its role, you can better interpret the order and structure within algebraic expressions.