Problem 70 - 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. $$ 1 \oplus(0 \oplus 1) $$

Step-by-Step Solution

Answer

Answer: The result of the expression is \(0\).

Step 1: First operation: \(0 \oplus 1\)#

According to the XOR definition, we have \(0 \oplus 1 = 1\) because exactly one of the bits \(0\) and \(1\) is \(1\).#

Step 2: Second operation: \(1 \oplus (0 \oplus 1)\)#

Now we can replace the result of the first operation in the expression: \(1 \oplus (0 \oplus 1) = 1 \oplus 1\). Using the XOR definition again, we have \(1 \oplus 1 = 0\), as both bits are \(1\).# So, the final evaluated result is: $$ 1 \oplus (0 \oplus 1) = 0 $$

Key Concepts

Binary Operator

In the digital world, a binary operator takes two binary values (0 and 1) and performs a calculation. Binary operators include fundamental arithmetic like addition and subtraction, but they also encompass logical operations, which are essential in computer science and digital electronics.

Operators like AND, OR, and XOR belong to this category and are foundational in performing logic operations on binary numbers. When working with binary operators, it's key to understand that they operate on bits and can be applied to sequences of bits known as binary numbers. Each operation is conducted on corresponding pairs of bits from the two numbers involved.

For example, consider the bitwise XOR operation on binary numbers 1010 and 0110:

• The first pair of bits is 1 (from 1010) and 0 (from 0110), and their XOR is 1.
• The second pair is 0 (from 1010) and 1 (from 0110), and their XOR is 1.
• The third pair is 1 (from 1010) and 1 (from 0110), and their XOR is 0 since both bits are equal.
• The last pair is 0 (from 1010) and 0 (from 0110), and their XOR is 0 as well.

The result of this XOR operation is 1100. Understanding binary operators is fundamental to solving problems using binary logic.

Bitwise Exclusive OR

The bitwise exclusive OR, or XOR, is a binary operator that follows a simple rule: If the two bits being compared are different, it returns 1; if they are the same, it returns 0. Its symbol is often denoted by \(\oplus\). It compares bits in a side-by-side fashion, which makes it extremely useful in various digital computations, such as error detection and cryptography.

Here's an easy way to remember the XOR operation outcomes based on the bits compared:

• 1 XOR 0 = 1
• 0 XOR 1 = 1
• 1 XOR 1 = 0
• 0 XOR 0 = 0

As exemplified in the original exercise, when you need to evaluate \(1 \oplus (0 \oplus 1)\), you start by resolving the expression in the parentheses which is \(0 \oplus 1\), yielding 1. Following that, you perform \(1 \oplus 1\), which gives 0. The XOR operator is associative, meaning expressions can be evaluated in any order when they are composed entirely of XOR operations. By mastering the XOR operation, one can decipher more complex binary arithmetic required for deeper computer science applications.

Logic Gates

When discussing binary operators in the realm of electronics, logic gates are the physical manifestation of these binary operations. They are the building blocks of digital circuits, with each gate implementing a Boolean function. A gate processes one or more input signals to produce an output signal.

The XOR gate, in particular, is a digital logic gate that implements exclusivity — it outputs true, or 1, only when the inputs are different. Here's a simple way to visualize the behavior of an XOR gate:

• If the gate receives inputs 0 and 1, the output is 1.
• If the inputs are 1 and 0, the output remains 1.
• For inputs 1 and 1, or 0 and 0, the output is 0.

Understanding how XOR gates function is vital for digital systems where decisions are based on binary conditions. Logic gates form the foundation of more sophisticated circuits like multiplexers, demultiplexers, and even computers themselves. Grasping the concept of logic gates, including XOR, equips one to progress towards creating and analyzing complex digital systems.