Problem 69 - 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. $$ 1 \oplus(1 \oplus 1) $$

Step-by-Step Solution

Answer

Question: Evaluate the XOR binary operation on the expression: \(1 \oplus (1 \oplus 1)\). Answer: 1

Step 1: Evaluate the inner XOR operation

We have to find the result of the inner part of the expression which is \((1 \oplus 1)\). $$ 1 \oplus 1 = 0 $$ The XOR operation for the two identical bits (1 and 1) yields 0.

Step 2: Evaluate the outer XOR operation

Now, we will evaluate the entire expression using the result from step 1: $$ 1 \oplus (1 \oplus 1) = 1 \oplus 0 $$ $$ 1 \oplus 0 = 1 $$ The XOR operation for the given bits (1 and 0) yields 1. Therefore, the evaluation of the expression \(1 \oplus (1 \oplus 1)\) is 1.

Key Concepts

Understanding Binary Operators

Binary operators are foundational concepts in computer science and mathematics. As the name suggests, a binary operator requires two operands to perform its operation. The exclusive or (XOR) operation, denoted by the symbol \( \oplus \), is a perfect example of a binary operator, as it takes two binary inputs and gives a single binary output.

The functioning of the XOR operation adheres to a simple rule: it outputs '1' if and only if one of the inputs is '1', and the other is '0'. Therefore, when both inputs are the same, the result is '0'.

• If the inputs are (1, 0) or (0, 1), XOR outputs 1.
• If the inputs are (0, 0) or (1, 1), XOR outputs 0.

Understanding binary operators are crucial not just for computer science students but for anyone diving into logic, algorithms, and programming, as these operations form the backbone of logic-based decisions and computer arithmetic.

Logic Gates and their Operations

At the core of digital electronics lie logic gates, which are physical manifestations of binary operators in hardware. These gates perform logical operations on one or more binary inputs to produce a single output. The XOR gate is a prime example of logic gates executing the XOR operation. In the realm of logic gates, the XOR operation is particularly intriguing for it embodies the logical equivalence to the 'either/or' condition in human language.

Truth Table of an XOR Gate

A useful way to visualize the behavior of an XOR gate is through its truth table which outlines all possible input combinations and their corresponding outputs:

• Input A = 0, Input B = 0 | Output = 0
• Input A = 0, Input B = 1 | Output = 1
• Input A = 1, Input B = 0 | Output = 1
• Input A = 1, Input B = 1 | Output = 0

This characteristic of the XOR gate is particularly employed in electronic circuits for error checking and in algorithms where conditions for exclusivity are necessary.

Bitwise Operations in Computing

Bitwise operations are operations that act directly on the binary digits, or bits, of a number. These are quite important in low-level programming, where manipulation of data at the bit level is required for performance and direct hardware interaction. Similar to how arithmetic operations like addition and subtraction are performed on numbers, bitwise operations manipulate bits in binary numbers.

For example, the bitwise XOR operation is a binary operation that takes two equal-length binary representations and performs the logical XOR on each pair of corresponding bits. In the given exercise, this concept was illustrated using the expression \((1 \oplus (1 \oplus 1))\), with the inner XOR executed first, followed by the outer one. Such bitwise operations are instrumental in computer graphics, cryptography, and network communications, where binary data must be processed efficiently.

• Bitwise XOR of 1 and 1 gives 0
• Bitwise XOR of 1 and 0 gives 1

Understanding bitwise operations opens up opportunities for problem-solving in areas where detailed control over binary data is crucial.