Problem 73 - Discrete Mathematics with Applications

Question

Construct a derivation tree for each expression. $$\mathbf{a} *(\mathbf{b}+\mathbf{c} / \mathbf{d})$$

Step-by-Step Solution

Answer

The completed derivation tree for the given expression \(\mathbf{a} *(\mathbf{b}+\mathbf{c} / \mathbf{d})\) is: \[ \begin{array}{ccccc} & * & & & \\ \mathbf{a} & & + & & \\ & & \mathbf{b}& / &\\ & & &\mathbf{c}&\mathbf{d} \end{array}_{} \] This tree visually represents the order of operations in the expression.

Step 1: Identify the order of operations

The order of operations is as follows: 1. Parentheses 2. Exponents 3. Multiplication and Division (from left to right) 4. Addition and Subtraction (from left to right) In this expression, we have: - parentheses: \((\mathbf{b}+\mathbf{c} / \mathbf{d})\) - multiplication: \(\mathbf{a} * ...\) - addition: \(\mathbf{b}+\mathbf{c} \) - division: \(\mathbf{c} / \mathbf{d} \) Now, we construct the derivation tree based on the order of operations.

Step 2: Begin the derivation tree

Start with the outermost operation, which is multiplication, as the root of the tree: \[ \begin{array}{c} * \\ \end{array}_{} \] Then add the two operands, \(\mathbf{a}\) and \((\mathbf{b}+\mathbf{c} / \mathbf{d})\).

Step 3: Add operands to the tree

Add the operand \(\mathbf{a}\) to the left side of the tree and parenthesized expression \((\mathbf{b}+\mathbf{c} / \mathbf{d})\) to the right side: \[ \begin{array}{ccc} & * & \\ \mathbf{a} & & (\mathbf{b}+\mathbf{c}/\mathbf{d} ) \end{array}_{} \]

Step 4: Break down the parenthesized expression

Inside the parentheses, we have an addition operation involving \(\mathbf{b}\) and the division operation \(\mathbf{c}/\mathbf{d}\). Add the '+' node to the right subtree: \[ \begin{array}{ccc} & * & \\ \mathbf{a} & & + \\ \end{array}_{} \] Now add \(\mathbf{b}\) on the left side and the division operation on the right side of the '+' node: \[ \begin{array}{ccccc} & * & & & \\ \mathbf{a} & & + & & / \\ & & \mathbf{b}& \mathbf{c} & \mathbf{d} \end{array}_{} \]

Step 5: Completed derivation tree

The completed derivation tree for the given expression \(\mathbf{a} *(\mathbf{b}+\mathbf{c} / \mathbf{d})\) is as follows: \[ \begin{array}{ccccc} & * & & & \\ \mathbf{a} & & + & & \\ & & \mathbf{b}& / &\\ & & &\mathbf{c}&\mathbf{d} \end{array}_{} \] This derivation tree visually represents the order of operations in the given expression.

Key Concepts

Order of Operations

Understanding the order of operations is essential when working with mathematical expressions. The order dictates the sequence in which operations should be performed to arrive at the correct result. This sequence avoids ambiguity and ensures consistency in solving mathematical problems.

• Parentheses: Operations within parentheses are performed first.
• Exponents: Following parentheses, any exponents (powers) are calculated next.
• Multiplication and Division: These operations are handled from left to right, after exponents.
• Addition and Subtraction: Finally, these are performed last, also from left to right.

This set of rules is commonly abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). It provides a systematic approach to simplify and solve expressions like a roadmap, ensuring the math operations align with established standards.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, operations, and sometimes parentheses or other grouping symbols. They represent specific numerical values or relationships. In the case of the given expression, a * \( (b+c/d) \),you see how various operations and parentheses come together to form a solvable query.
Here are some elements that make up mathematical expressions:

• Variables: Symbols like \(a, b, c,\) and \(d\) represent unknown or changeable values.
• Numbers: These are constant values used with variables.
• Operators: Symbols such as \(+, -, *, /\) indicate the operations to be performed.
• Parentheses: Group parts of the expression to indicate priority in calculations.

Understanding this structure helps in evaluating or simplifying expressions, ensuring that each part is handled correctly following the prescribed order of operations.

Multiplication and Division

Multiplication and division are two fundamental operations that often occur together. In the context of the order of operations, these are performed sequentially from left to right after handling any parentheses and exponents. In the given expression, multiplication and division play significant roles in determining the outcome.For example:

• Multiplication: Involves repetitive addition of a number. It is used to calculate the product of two numbers, such as \(a \times (b+c/d)\).
• Division: Splits a number into equal parts or determines how many times one number is contained within another. It is seen in the expression as \((c/d)\).

Being attentive to these operations when building or analyzing derivation trees helps to correctly visualize and solve mathematical expressions.

Addition and Subtraction

Addition and subtraction are basic arithmetic operations that are usually solved after multiplication and division according to the order of operations. These operations help combine or reduce quantities in mathematical expressions.In the expression given, subtraction does not appear, but addition is evident:

• Addition: Combines two or more numbers or variables to provide their total. For instance, in the expression, we see \(b+c\) happening inside the parentheses.
• Subtraction: Though not present in this example, it would similarly follow the order of operations being handled after any grouping symbols and the previous operations.

Grasping these operations and where they fit within expressions enables students to correctly set up their calculations, ensuring accuracy, especially in more complex derivations.