Problem 80 - 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. $$ x \oplus y=(x+y)(x y)^{\prime} $$

Step-by-Step Solution

Answer

The proof can be demonstrated by enumerating all possible values of x and y, computing \( x \oplus y \) and \( (x+y)(x y)^{\prime} \) for these combinations, and comparing the results. After evaluating, we find that the results match for all combinations of x and y. Therefore, the given equation \( x \oplus y = (x+y)(x y)^{\prime} \) is proved true.

Step 1: Enumerate all possibilities for x and y

Start by enumerating all possible values of x and y. In binary terms, x and y can be either 0 or 1. This gives us four possible combinations: (0, 0), (0, 1), (1, 0), and (1, 1).

Step 2: Compute x XOR y for all possibilities

Next, compute the value of \( x \oplus y \) for each of the combinations as given by its definition. Write down these results as well.

Step 3: Computing (x+y)(xy)’ for all possibilities

Now, compute the value of \( (x+y)(x y)^{\prime} \) for each of the four combinations. According to the definition, \( (x+y) \) denotes the logical OR operation and \( (x y)^{\prime} \) denotes logical AND operation followed by a NOT operation.

Step 4: Comparing the two columns

Finally, compare the results from Step 2 and Step 3. If they match for all combinations of x and y, then the proof is completed. If they do not, then there is a mistake somewhere.

Step 5: Conclusion

The conclusion should restate the fact that we have shown that for all possible values of x and y, the value of \( x \oplus y \) is equal to \( (x+y)( x y)^{\prime} \). So, the given equation is proved true.

Key Concepts

Binary Operations

Binary operations are fundamental in discrete mathematics and computer science. They operate on two inputs to produce a single output. In binary operations, both inputs are typically binary digits, 0 or 1.
This concept is crucial as it forms the basis for various operations in digital electronics and logic gates.

• Binary operations include addition, subtraction, multiplication, and logical operations such as AND, OR, and XOR.
• Each operation follows a set of rules or definitions that describe the output for given inputs.
• They are essential for tasks involving computation and data manipulation in binary form.

Understanding binary operations allows you to decode how complex systems, like computers, execute calculations and process information.

XOR (Exclusive OR)

XOR, or Exclusive OR, is a special binary operation denoted by the symbol \( \oplus \). Unlike the regular OR operation, XOR only returns true or 1 when the inputs differ, meaning exactly one of them is 1. It is a common operation used in digital logic and computer architecture.

• For XOR, the significant characteristic is its binary nature, focusing on the differences between the bits.
• The result is 1 if one, and only one, of the inputs is 1. Otherwise, it results in 0.
• This operation is particularly valuable in error detection and correction schemes and cryptographic algorithms.

The XOR operation is sometimes referred to as 'modulo-2 addition' because of its similarity to addition under mod 2 operations, highlighting its utility in digital circuits.

Truth Tables

Truth tables are a tool used to represent the output of logical operations for all possible input combinations. They are particularly helpful in simplifying complex logical expressions and proving logical equivalences, such as the one between \( x \oplus y \) and \( (x+y)(xy)' \).
A truth table outlines all possible combinations of the inputs and the corresponding outputs.

• Each row of the table represents a unique combination of inputs.
• They help in visualizing how the logic operation transforms the inputs into the output.
• Truth tables provide a systematic approach to verify logical equivalences and laws.

By analyzing truth tables, you can observe patterns and identify errors, making them an invaluable resource in both theoretical and practical applications.

Logical Operators

Logical operators, like AND, OR, and NOT, perform logical operations on binary values. They are the building blocks for creating complex logical conditions and expressions.

• AND (\( \land \)) results in true if both operands are true.
• OR (\( \lor \)) results in true if at least one operand is true.
• NOT (\( \lnot \)) inverts the value of a single operand.

Understanding these basic logical operators is essential for constructing more complex expressions and computing various logical outcomes. When combined, these operators can express any logical statement or condition used in programming, digital circuits, and mathematical proofs.
Understanding how these operators work contributes to problem-solving and reasoning within discrete mathematics.