Problem 9 - Discrete Mathematics with Applications

Question

Using the BCD expansions of the decimal numbers 0 through 9, develop a boolean function \(f(w, x, y, z)\) such that the decimal value of \((w x y z)_{\text {two }}\) is divisible by: Two

Step-by-Step Solution

Answer

The boolean function f(w, x, y, z) can be defined as \(f(w, x, y, z) = \overline{z}\), where the NOT operation is represented by the overline. This function checks if the last digit z of the binary number (wxyz)_2 is 0, which indicates that the number is divisible by 2.

Step 1: Check the value of z

Since a binary number is divisible by 2 when z is 0, we need to determine the value of z in our boolean function. Step 2: Create the boolean function

Step 2: Define the boolean function f(w, x, y, z)

Since we only need to check if z is 0, our boolean function will be f(w, x, y, z) = NOT(z), where NOT is a binary logical operation that inverts the given input. If z = 0, the output f(w, x, y, z) will be 1, indicating that the binary number (wxyz)_2 is divisible by 2. If z = 1, the output will be 0, indicating that the binary number is not divisible by 2.

Key Concepts

BCD (Binary-Coded Decimal)

Binary-Coded Decimal (BCD) is a way to express each digit of a decimal number in binary form. This is different from converting the entire number into binary. Each decimal digit, from 0 to 9, is represented as a four-bit binary equivalent. For example, the decimal number '9' is represented in BCD as '1001'. The BCD format makes it easier to handle numerical data in digital systems that operate in a base-10 environment, particularly in electronics like calculators.

• Simplified conversion: Each decimal digit directly converts to a fixed binary sequence, reducing conversion complexity.
• Readability: It allows easier readability and conversion between human-readable numbers and machine representations.

It's important to know that while BCD is helpful in certain digital applications, it can be slightly inefficient compared to pure binary numbers in terms of storage, as it uses more bits to represent the same number.

Divisibility rules

Divisibility rules are guidelines that help us determine if one number can be divided by another without leaving a remainder. The most common rules apply to numbers like 2, 3, 5, and 10, each having its own rule. For divisibility by 2, any number must have its last digit as even. In binary terms, this means that the least significant bit (LSB) should be 0. This is why, in the boolean function from our exercise, checking if z (the LSB) is 0 is crucial.

• Practicality: These rules simplify large number calculations without having to perform full division.
• Quick checks: For binary numbers, easily check and prove their divisibility by examining the rightmost bits.

Understanding and applying divisibility rules can significantly speed up solving numeric problems, especially those involving checks for division by small numbers.

Binary numbers

Binary numbers are the cornerstone of modern computing systems. Unlike our customary decimal system, which is base-10, the binary system is base-2, using only two digits: 0 and 1. Everything a computer processes is inherently binary. The values we input, process, and output are in this form, making understanding binary crucial for anyone who wants to delve deeper into computing. In binary, place value plays a significant role, just like in decimal, but based on powers of 2.

• Efficiency: Binary is efficient for electronic devices since it aligns with digital circuit operation, using transistors that have two states: on (1) and off (0).
• Simplicity: The simplicity of binary makes it a natural fit for logical operations and boolean functions.

Binary numbers provide the ability to perform complex operations with simple off-on logic, forming the basis for digital signal processing and computational logic.

Logic gates

Logic gates are the fundamental building blocks of digital circuits, which help process binary data by performing basic logical functions. Each gate implements a specific logical operation, such as AND, OR, NOT, NAND, NOR, XOR, and XNOR. These operations correspond to various combinations of binary inputs leading to specific outputs based on logical rules. For instance, in our boolean function from the exercise, we use the NOT gate to check the divisibility by 2 logic by inverting the input 'z'. If z = 0, NOT(z) yields 1, indicating divisibility, and vice-versa.

• Core Operations: Operations like AND, OR, and NOT are the basic building blocks that can be combined to form complex circuits.
• Truth Tables: Each gate operation can be described using a truth table that shows all combinations of input-output states.

Understanding how these gates manipulate binary inputs helps to comprehend more complex digital systems, allowing for the design of intricate circuits right from simple logical computations to sophisticated processing units.