Problem 70 - Discrete Mathematics with Applications

Question

Determine if each is a legal expression. $$-a * b / c+d$$

Step-by-Step Solution

Answer

The expression \(-a * b / c + d\) is a legal expression, as it follows the conventions of mathematics, has no ambiguities, and all variables and operators are used correctly.

Step 1: Examine the given expression

Let's take a look at the expression provided: \(-a * b / c + d\).

Step 2: Check for appropriate use of operators

The expression contains multiplication (\(*\)), division (\(/\)), and addition (\(+\)) operators. These operators are being used correctly between variables or constants. There are no instances of two operators being used side by side, which would be illegal.

Step 3: Verify the correct usage of variables and constants in the expression

All variables (a, b, c, d) are used correctly in the expression, with no illegal double variables or constants, such as \(aa\), \(2a\), or \(22\).

Step 4: Determine the expression's ambiguity

Due to the Order of Operations (PEMDAS/BODMAS) used in mathematics, the expression is not ambiguous. The operations would be executed in the following order: multiplication and division (from left to right), and finally addition.

Step 5: Conclusion

The expression \(-a * b / c + d\) is a legal expression, as it follows the conventions of mathematics, has no ambiguities, and all variables and operators are used correctly.

Key Concepts

Order of Operations

In mathematics, the Order of Operations is a set of rules that dictates the correct sequence in which operations should be performed in a mathematical expression. This helps avoid any confusion or ambiguity in calculations. A common acronym used to remember this order is PEMDAS (or BODMAS in some regions). This stands for:

• Parentheses (Brackets): Solve anything in parentheses first.
• Exponents (Orders): Calculate powers or square roots next.
• Multiplication and Division: Perform these operations from left to right.
• Addition and Subtraction: Finally, add or subtract from left to right.

For example, in the expression \(-a * b / c + d\), multiplication and division are prioritized over addition due to this order, ensuring the expression is calculated properly.
Following these guidelines avoids errors and ensures consistency in solving mathematical problems.
Understanding the Order of Operations is fundamental for solving equations accurately.

Legal Expressions

A legal expression in mathematics is one that is structured according to the accepted rules and syntax, without any contradictions or incorrect usage of operations and symbols.
This means there are no misplaced operators, no illegal combinations of variables or constants, and each mathematical operation is performed in a valid sequence.
When analyzing an expression like \(-a * b / c + d\), it is important to ensure that:

• Operators are correctly placed and used between numbers or variables.
• Variables and constants do not form illegal groupings (like repeated variables without operations).
• The overall expression conforms to mathematical norms regarding precedence and sequence.

Confirming an expression is legal ensures it can be processed correctly without ambiguity, leading to a valid and reliable result.

Arithmetic Operators

Arithmetic Operators are the basic symbols used in mathematics to perform calculations.
These include addition (\(+\)), subtraction (\(-\)), multiplication (\(*\)), and division (\(/\)). Each operates on numbers or variables to yield a new value.
In the expression \(-a * b / c + d\):

• Multiplication (\(*\)): This operator is used to multiply two numbers or variables.
• Division (\(/\)): This splits one number by another.
• Addition (\(+\)): Adds one value to another.
• Subtraction (\(-\)): Although implicit here as a negative sign, it’s an operator that deducts value.

Recognizing how these operators interact is crucial for solving mathematical expressions. Each operator has specific rules for usage, emphasizing the importance of understanding their function and order in an expression.

Mathematical Syntax

Mathematical Syntax refers to the formal set of rules that define the structure and composition of mathematical expressions.
It includes the correct placement and usage of numbers, operators, variables, and symbols to ensure that an expression is mathematically meaningful and can be interpreted unambiguously.
In expressions such as \(-a * b / c + d\):

• The arrangement of operations follows mathematical syntax norms.
• Variables and numbers are properly placed without errors.
• Each element of the expression contributes to a cohesive structure, ensuring accurate computation.

Adhering to proper syntax is like following grammar rules in a language; it makes the difference between an expression that conveys the intended meaning and a jumble of symbols that cannot be deciphered or solved correctly. Understanding and applying these rules is key in mathematics.