Problem 12 - Discrete Mathematics with Applications

Question

Construct a logic table for each boolean function defined by each boolean expression. $$\left(x+y^{\prime}+z\right)\left(x^{\prime}+y+z^{\prime}\right)$$

Step-by-Step Solution

Answer

The logic table for the boolean expression \((x+y'+z)(x'+y+z')\) is given as follows: | \(x\) | \(y\) | \(z\) | \((x+y'+z)(x'+y+z')\) | |-------|-------|-------|-----------------------| | 0 | 0 | 0 | 1 | | 0 | 0 | 1 | 1 | | 0 | 1 | 0 | 0 | | 0 | 1 | 1 | 1 | | 1 | 0 | 0 | 1 | | 1 | 0 | 1 | 0 | | 1 | 1 | 0 | 1 | | 1 | 1 | 1 | 1 |

Step 1: Identify the variables in the boolean expression

In the given boolean expression \((x+y'+z)(x'+y+z')\), the variables involved are \(x\), \(y\), and \(z\).

Step 2: List all possible input combinations

Since we have 3 variables, we have \(2^3 = 8\) possible input combinations. These combinations are: - 000 - 001 - 010 - 011 - 100 - 101 - 110 - 111

Step 3: Evaluate the boolean expression for each input combination

Now, for each combination, evaluate the given boolean expression \((x+y'+z)(x'+y+z')\): 1. For \(x=0\), \(y=0\), and \(z=0\), the expression becomes \((0+1+0)(1+0+1) = (1)(2) = 1\). 2. For \(x=0\), \(y=0\), and \(z=1\), the expression becomes \((0+1+1)(1+0+0) = (2)(1) = 1\). 3. For \(x=0\), \(y=1\), and \(z=0\), the expression becomes \((0+0+0)(1+1+1) = (0)(3) = 0\). 4. For \(x=0\), \(y=1\), and \(z=1\), the expression becomes \((0+0+1)(1+1+0) = (1)(2) = 1\). 5. For \(x=1\), \(y=0\), and \(z=0\), the expression becomes \((1+1+0)(0+0+1) = (2)(1) = 1\). 6. For \(x=1\), \(y=0\), and \(z=1\), the expression becomes \((1+1+1)(0+0+0) = (3)(0) = 0\). 7. For \(x=1\), \(y=1\), and \(z=0\), the expression becomes \((1+0+0)(0+1+1) = (1)(2) = 1\). 8. For \(x=1\), \(y=1\), and \(z=1\), the expression becomes \((1+0+1)(0+1+0) = (2)(1) = 1\).

Step 4: Construct the logic table

Based on the evaluations in Step 3, we can now construct the logic table for the given boolean expression. The table is as follows: | \(x\) | \(y\) | \(z\) | \((x+y'+z)(x'+y+z')\) | |-------|-------|-------|-----------------------| | 0 | 0 | 0 | 1 | | 0 | 0 | 1 | 1 | | 0 | 1 | 0 | 0 | | 0 | 1 | 1 | 1 | | 1 | 0 | 0 | 1 | | 1 | 0 | 1 | 0 | | 1 | 1 | 0 | 1 | | 1 | 1 | 1 | 1 | And that's the logic table for the given boolean expression!

Key Concepts

Boolean Algebra

Boolean algebra is a branch of mathematics that deals with operations on logical values and incorporates the principles of logic. It is named after George Boole, who first defined an algebraic system of logic in the mid-19th century. In the context of computing and electronics, boolean algebra is used to analyze and simplify digital circuits.

Boolean algebra involves variables that can have two possible values: true (1) or false (0). Operations like AND (conjunction), OR (disjunction), NOT (negation), NAND (not-and), NOR (not-or), XOR (exclusive-or), and XNOR (equivalence) are the basic building blocks connecting these values. For example, in the exercise given, the expression \( (x+y'+z)(x'+y+z') \) can be translated to using AND, OR, and NOT operations. These operations are represented by the symbols \( + \) for OR, \( ' \) for NOT, and the absence of a symbol for AND.

In simplifying a boolean expression like the one provided, it is important to apply the laws of boolean algebra. These laws, such as commutativity, associativity, distribution, and DeMorgan's theorem, allow for the transformation of complex expressions into simpler forms, which is essential for optimizing logic circuits in digital electronics.

Truth Table

A truth table is a mathematical table used in logic—specifically in connection with boolean algebra, boolean functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments. In essence, it's a tabular representation of all possible values of a logic expression based on the input values.

For each boolean variable, the table has two rows: one for true (1) and one for false (0). With each additional variable, the size of the truth table doubles. Therefore, for three variables, the truth table will have \(2^3 = 8\) rows, representing all possible combinations of truth values for the variables involved. Students should note that truth tables are a fundamental tool for visualizing and checking the outputs of boolean expressions, providing a clear means to infer the functioning of logic circuits. The step-by-step solution demonstrated the construction of a truth table by evaluating the boolean expression with every possible combination of input values.

Logic Gates

Logic gates are physical devices that implement boolean functions, that is, they perform a logical operation on one or more binary inputs and produce a single binary output. In digital circuits, logic gates are the building blocks that carry out the operations laid out by boolean algebra. The basic types of logic gates include AND, OR, NOT, NAND, NOR, XOR, and XNOR.

Each gate has a symbolic representation and truth table that delineates its behavior. For example, an AND gate performs the multiplication of input signals, an OR gate corresponds to the addition, and a NOT gate inverts its input. In our exercise to construct a logic table for the boolean expression \( (x+y'+z)(x'+y+z') \), logic gates would translate this into a real-world circuit with perhaps two AND gates for multiplication, three OR gates for addition, and two NOT gates to invert \( y \) and \( z \) respectively.

Understanding how these gates interact in a circuit to produce the desired output is crucial for the design and analysis of digital systems. By using truth tables to predict the outcomes of different gate combinations, students can gain insight into how complex digital circuits are put together and function.