Problem 1 - Discrete Mathematics with Applications

Question

Evaluate the boolean expression \(x\left(y z^{\prime}+y^{\prime} z\right)\) at the ordered triplets (1,0,1) and (1,1,1)

Step-by-Step Solution

Answer

Answer: The given boolean expression evaluates to 1 for the ordered triplet (1,0,1) and evaluates to 0 for the ordered triplet (1,1,1).

Step 1: Write down the given expression

The boolean expression is \(x(yz'+y'z)\). We will evaluate this expression at the given ordered triplets \((1,0,1)\) and \((1,1,1)\).

Step 2: Evaluate the expression at (1,0,1)

We substitute x=1, y=0, and z=1 in the given expression: \(x(yz'+y'z) = 1(0 \cdot 1^{\prime} + 0^{\prime} \cdot 1)\) Now, apply NOT (') operation, note that \(1^{\prime}=0\) and \(0^{\prime}=1\): \(= 1(0 \cdot 0 + 1 \cdot 1)\) Use AND operation (multiply terms within brackets): \(= 1(0 + 1)\) Now, apply OR operation (add terms within brackets): \(= 1\) So, the boolean expression evaluates to 1 for the ordered triplet \((1,0,1)\).

Step 3: Evaluate the expression at (1,1,1)

Now, let's substitute x=1, y=1, and z=1 in the given expression: \(x(yz'+y'z) = 1(1 \cdot 1^{\prime} + 1^{\prime} \cdot 1)\) Apply NOT (') operation: \(= 1(1 \cdot 0 + 0 \cdot 1)\) Use AND operation (multiply terms within brackets): \(= 1(0 + 0)\) Now, apply OR operation (add terms within brackets): \(= 1(0)\) Finally, use AND operation (multiply): \(= 0\) So, the boolean expression evaluates to 0 for the ordered triplet \((1,1,1)\).

Step 4: Final answer

After evaluating the given boolean expression at the specific ordered triplets, we found: - For \((1,0,1)\), the expression evaluates to 1. - For \((1,1,1)\), the expression evaluates to 0.

Key Concepts

Boolean Algebra

Boolean algebra is a branch of mathematics that deals with operations on logical values and incorporates binary variables. It operates under a set of rules quite different from the algebra of numbers. Boolean algebra is named after George Boole, a mathematician who first defined an algebraic system for logic in the 19th century.

In Boolean algebra, the variables are generally assigned a value of either 0 or 1, where 0 typically represents the logical state 'false' and 1 represents 'true'. The basic operations in Boolean algebra include AND (conjunction, represented by a dot or multiplication), OR (disjunction, represented by a plus or addition), and NOT (negation, often denoted by an apostrophe or prime symbol).

Understanding this concept not only forms the basis for evaluating Boolean expressions as in the given exercise but also has practical application in the field of computer science, particularly in the design of circuits and digital systems.

Truth Table

A truth table is a mathematical table used to determine the output of a Boolean expression based on all possible inputs. It lists all possible combinations of input values, and for each combination, it shows the result of applying the logical operations indicated by the Boolean expression.

To build a truth table, you must identify all input variables, list all possible combinations of their values, and specify the outputs from the Boolean operations. The truth table thus serves as a visual representation of how logic gates will behave in different scenarios, making it an indispensable tool in both learning Boolean algebra and in practical applications such as troubleshooting logical circuits.

Logical Operations

Logical operations are the actions that we perform on Boolean values in Boolean algebra. The fundamental logical operations include:

• AND (also notated as multiplication): The output is true if and only if all inputs are true.
• OR (also notated as addition): The output is true if at least one of the inputs is true.
• NOT (also notated with a prime symbol as negation): It inverts the value of the input, turning true to false and vice-versa.

In the given exercise, these operations are employed to evaluate the expression at given points. The step-by-step solution demonstrates how to substitute variables with values and apply the logical operations to obtain the final result. This process is crucial for verifying the correctness of logical circuits and programming logical statements in computer science.