Problem 79 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

Question: Prove that the XOR operator is commutative. Answer: By showing the commutativity of the XOR operator for all possible combinations of bits x and y using a truth table, we can conclude that the XOR operator \(\oplus\) is commutative: \(x \oplus y = y \oplus x\).

Step 1: Create a truth table for XOR operator

Let us create a truth table for the XOR operator, showing all possible combinations of bits x and y: \(x\) | \(y\) | \(x \oplus y\) --- | --- | ------ 0 | 0 | 0 0 | 1 | 1 1 | 0 | 1 1 | 1 | 0

Step 2: Show commutativity for all possible combinations of bits x and y

Now let's demonstrate commutativity of the XOR operator for all possible combinations of x and y: 1. If x = 0 and y = 0, then \(x \oplus y = y \oplus x = 0 \oplus 0 = 0\). The XOR operator is commutative in this case. 2. If x = 0 and y = 1, then \(x \oplus y = 0 \oplus 1 = 1\), and \(y \oplus x = 1 \oplus 0 = 1\). The XOR operator is also commutative in this case. 3. If x = 1 and y = 0, then \(x \oplus y = 1 \oplus 0 = 1\), and \(y \oplus x = 0 \oplus 1 = 1\). The XOR operator is commutative in this case, too. 4. If x = 1 and y = 1, then \(x \oplus y = 1 \oplus 1 = 0\), and \(y \oplus x = 1 \oplus 1 = 0\). The XOR operator is commutative in this case as well.

Step 3: Conclude that XOR operator is commutative

Since we have shown that the XOR operator \(\oplus\) is commutative for all possible combinations of bits x and y, we can conclude that: $$x \oplus y = y \oplus x$$

Key Concepts

Binary Operators

Binary operators are fundamental concepts in computer science and mathematics that take two operands and perform a specific operation on them. The result of this operation is typically a single value. Common binary operators include addition (+), subtraction (-), multiplication (*), and division (/). In the realm of computer science, we also encounter binary logical operators such as AND, OR, XOR (exclusive OR), and NOT, which work on binary digits (bits).

These operators are integral to programming and digital circuit design as they lay the foundation for decision-making and data manipulation. While arithmetic operators work with numeric values, logical binary operators are primarily used in truth-functional logic tasks, handling true or false values, commonly represented as 1 and 0 in binary systems.

Truth Tables

Truth tables are a visual representation used in logic and computing to depict all the possible outcomes of a logical operation. Each row in the truth table corresponds to a unique combination of input values, and the resulting column illustrates the output for that combination. It's like a map that guides us through the landscape of logical functions.

For binary operators, truth tables will typically have two input columns representing the two variables or operands involved, and one output column showing the result of the operation. They serve as an excellent tool for understanding logical operations, such as the XOR operator, by providing a clear and comprehensive overview of how the operator behaves for every possible input.

Bitwise Operations

Bitwise operations are powerful tools in computer programming that perform operations on individual bits within a binary number. Think of them like precision maneuvers that target the tiniest elements in computing: the bits. Operators like AND, OR, XOR, and NOT can be applied bitwise to perform binary arithmetic, manipulate flags, and control binary states in software development.

The XOR operator, or exclusive OR, comes under scrutiny when discussing bitwise operations, as it has some unique properties compared to its logical counterparts. It sets the resulting bit to 1 only if one and only one of the corresponding bits of the operands is 1. Bitwise operations, such as XOR, are commonly used in cryptographic algorithms, error detection, and parity checks.

Commutative Property

The commutative property is a basic rule in mathematics that applies to certain operations, stating that changing the order of the operands does not change the result. This rule is familiar to most people through elementary arithmetic, where we learn that 2+3 is the same as 3+2. When it comes to binary operations, if an operator is commutative, you can switch the operands around, and you'll end up with the same outcome.

In the case of the XOR operator, the commutative property comes into play, allowing us to affirm that \(x \oplus y = y \oplus x\). This characteristic is essential in simplifying complex logical expressions and is particularly useful in various fields, such as computer programming and digital logic design, where flexibility in the order of operations can lead to more efficient algorithms and clearer code.