Problem 7 - Discrete Mathematics with Applications

Question

Determine if each is a boolean expression, where each variable is boolean. $$ \left(x y+y^{\prime} z^{\prime}\right)^{\prime} $$

Step-by-Step Solution

Answer

The given expression, \(\left(x y+y^{\prime} z^{\prime}\right)^{\prime}\), contains boolean variables \(x, y, y', z, z'\) and boolean operators AND (\(\cdot\)), OR (\(+\)), and NOT (\('\)). It satisfies the boolean algebra rules with the correct application of these operators on the variables. Hence, it is a boolean expression.

Step 1: Identify the Variables

Let's identify the boolean variables in the given expression: \( x, y, y', z, z' \)

Step 2: Identify the Operators

Now let's identify the boolean operators in the given expression: AND operator (\(\cdot\)): between x and y, and also between \(y'\) and \(z'\) OR operator (\(+\)): between the two terms NOT operator (\('\)): on y and z

Step 3: Check the Boolean Rules

Let's check if the given expression satisfies the boolean algebra rules using the identified variables and operators: The AND operator is applied correctly between x and y, and between \(y'\) and \(z'\) since it operates on two boolean variables. The OR operator is correctly applied between the two terms, combining the results of the AND operations, which are also boolean expressions. The NOT operator is also applied correctly to y and z, as it operates on a single boolean variable.

Step 4: Conclusion

The given expression, \(\left(x y+y^{\prime} z^{\prime}\right)^{\prime}\), satisfies all the boolean algebra rules and has the correct application of boolean operators to boolean variables. Therefore, it is a boolean expression.

Key Concepts

Boolean Variables

In the realm of digital logic and computer science, boolean variables are the building blocks of boolean expressions. They are the elements that store boolean values, which are the classic binary choices: true or false, often represented as 1 and 0 respectively.

In an exercise, identifying boolean variables serves as the first step to understand the overall structure of the boolean expression. A variable like 'x' in a boolean context will have only two possible values, 1 (true) or 0 (false).

• When you see a variable such as 'y', assume it can either be 1 or 0.
• A variable followed by a prime symbol, such as 'y'^', indicates the negation or the logical NOT of that variable, flipping its value from 1 to 0 or vice versa.

This binary characteristic of variables is crucial because the entire foundation of boolean logic hinges on binary states.

Boolean Operators

Connecting these boolean variables are boolean operators, which are the tools we use to build more complex boolean expressions. The primary operators are AND, OR, and NOT, each performing a specific logical function.

The AND operator, denoted by a dot (\( \bullet \)) or simply by concatenating the variables (xy), returns true only if both operands are true. It's like a logical 'all conditions must be met.'

The OR operator, represented by a plus symbol (\( + \)), is inclusive and returns true if at least one operand is true, acting as a 'one or the other or both' condition.

The NOT operator, denoted by a prime symbol (\( ' \)), inverts the value, transforming a true into a false and vice versa. Sometimes, it is also represented by an overbar ( \bar{x} ).

Understanding how these operators work individually and within expressions enables students to deconstruct complex logical problems into simple, solvable parts.

Boolean Algebra Rules

Just as in conventional algebra, there are rules to follow in boolean algebra, formulated to manipulate boolean expressions reliably. These rules ensure that expressions simplify correctly, ultimately leading to the creation of optimized logical structures.

Key boolean algebra rules include the Identity Law, Null Law, Complement Law, and Idempotent Law:

Key Boolean Algebra Rules

• Identity Law: States that any variable ANDed with 1 or ORed with 0 will result in the variable itself, e.g., x • 1 = x, x + 0 = x.
• Null Law: Declares that a variable ANDed with 0 always gives 0, or ORed with 1 always results in 1, e.g., x • 0 = 0, x + 1 = 1.
• Complement Law: Indicates that a variable ANDed with its complement (NOT) is always 0, and ORed with its complement is always 1, e.g., x • x' = 0, x + x' = 1.
• Idempotent Law: Means that a variable ANDed or ORed with itself equals itself, e.g., x • x = x, x + x = x.

Grasping these rules empowers students to validate and simplify boolean expressions, much like the step-by-step solution process of identifying a proper boolean expression. Sticking to these rules is a foolproof way to handle boolean algebra and guarantees the correctness of logical operations.