Problem 72 - Discrete Mathematics with Applications

Question

Use the following definition of the binary operator \(\mathrm{XOR}\) , denoted by \(\oplus,\) for Exercises \(69-81 .\) $$ x \oplus y=\left\\{\begin{array}{ll}{1} & {\text { if exactly one of the bits } x \text { and } y \text { is } 1} \\ {0} & {\text { otherwise }}\end{array}\right. $$ Evaluate each. $$ 1 \uparrow(0 \oplus 1) $$

Step-by-Step Solution

Answer

The short answer is: Evaluate the expression \(1 \uparrow(0 \oplus 1)\) using the given definition of XOR. First, we find \(0 \oplus 1 = 1\), as exactly one of the bits is 1. Now, we can evaluate the expression as \(1 \uparrow(0 \oplus 1) = 1 \uparrow 1 = 1\). Thus, the expression evaluates to \(1\).

Step 1: Calculate (0 ⊕ 1)

According to the given definition of the XOR operation, we have: $$ 0 \oplus 1 = \left\\{ \begin{array}{ll}{1} & \text{ if exactly one of the bits 0 and 1 is 1}\\ {0} & \text{ otherwise }\end{array}\right. $$ Since exactly one of the bits (1) is 1, we get: $$ 0 \oplus 1 = 1 $$

Step 2: Evaluate 1 ^ (0 ⊕ 1)

Now that we know \((0 \oplus 1) = 1\), we can evaluate the expression: $$ 1 \uparrow(0 \oplus 1) = 1 \uparrow 1 = 1 $$ Since \(1 \uparrow 1 = 1\), the expression evaluates to 1 .

Key Concepts

Bitwise Operations

Bitwise operations are a key part of computer science and mathematics, allowing manipulation of data at the binary level. The basic premise of a bitwise operation is that it operates on the bits of binary numbers directly. This is particularly useful in scenarios where performance is crucial, as these operations are typically executed very fast by the processor.

One of the fundamental bitwise operations is XOR, represented by the symbol \( \oplus \). XOR works by comparing each corresponding pair of bits in two numbers and applying the following rule:

• If one bit is 1 and the other is 0, the XOR result is 1.
• If both bits are the same, the XOR result is 0.

This logical rule is useful in various applications, such as error detection and encryption. By understanding the rules of bitwise operations, you can manipulate binary data effectively to achieve desired computational results.

Discrete Mathematics

Discrete mathematics involves studying mathematical structures that are fundamentally discrete rather than continuous. These structures are essential in a variety of fields, including computer science, cryptography, and logic.

In the realm of discrete mathematics, logical operations, and bitwise calculations, such as XOR, play a critical role. This is largely because discrete mathematics often deals with distinct values and the laws of logic. For example, when studying combinations or networks, using logical operations lets us model and solve problems.

The XOR operation is a crucial building block in discrete mathematics. It is often used to represent certain types of logic circuits, commonly known as combinatorial logic. By understanding discrete structures and operations, you can better grasp how mathematical logic applies to computing and problem-solving.

Logical Operations

Logical operations form the backbone of both mathematics and computer science. These operations allow us to evaluate and simplify logical statements. A logical operator acts on one or more operand to produce a true or false result.

The XOR logical operation is unique among logical operators because it produces true (or 1) only when exactly one of its operands is true (or 1), not both. This property makes XOR suitable for various computational routines where exclusive choices occur. For example, XOR can effectively manage toggling operations in computer algorithms where a condition should change from true to false or vice versa.

Understanding logical operations, like XOR, allows us to create more efficient algorithms and systems. By applying these operations correctly, we can solve complex problems by breaking them down into simpler logical steps, ultimately leading to quantitative, correct solutions in programming and electronic logic design.