Problem 19 - Discrete Mathematics with Applications

Question

Mark each statement as true or false. $$x \uparrow y=y \uparrow x$$

Step-by-Step Solution

Answer

The given statement \(x \uparrow y = y \uparrow x\) or \(x^y = y^x\) is false. A counterexample is when x = 3 and y = 2, where \(3^2 \neq 2^3\).

Step 1: Identify notation

The given problem provides the following equation: \(x \uparrow y = y \uparrow x\). We can recognize that the \(\uparrow\) symbol represents exponentiation operation. So we can rewrite it as \(x^y = y^x\).

Step 2: Analyze the expression

In order to determine whether the expression \(x^y = y^x\) is true or false, we can evaluate the expression for different values of x and y. If there are counterexamples where the equation doesn't hold true, then we can conclude that it's false. Example 1: Let's take x = 2 and y = 4: \[2^4 = 16\] \[4^2 = 16\] In this particular case, the expression is true. Example 2: Let's take x = 3 and y = 2: \[3^2 = 9\] \[2^3 = 8\] In this case, the expression is false. Since we have found a counterexample where the given expression \(x^y = y^x\) doesn't hold true, we can conclude that the statement is false.

Key Concepts

Commutative Property

The commutative property is a fundamental principle in mathematics. It states that you can change the order of numbers in an operation without changing the result. The commutative property applies to addition (e.g., \(a + b = b + a\)) and multiplication (e.g., \(a \times b = b \times a\)). However, not all operations are commutative. This means changing order might change the result.
Exponentiation, as discussed in our exercise, is one such operation that isn’t commutative. Comparing the expressions \(x^y\) and \(y^x\), you might immediately realize that swapping the base and the exponent will generally result in different outcomes. While for certain values, such as \(x = 2\) and \(y = 4\), the result remains the same, this isn’t universally true. This lack of universal commutativity is precisely why exponentiation is non-commutative.

Counterexample

When exploring whether a statement is universally true in mathematics, providing a counterexample can be a powerful method for demonstrating that it's false. A counterexample is a specific case where the general statement doesn’t hold.
In the context of our exercise about exponentiation, we were testing if \(x^y = y^x\) is always true. By plugging in numbers like \(x = 3\) and \(y = 2\), we found that \(3^2 = 9\) and \(2^3 = 8\), so the statement fails here.

• Once a single counterexample is found, it’s enough to prove the general statement false.
• This method of using specific instances to show a statement doesn't hold is commonly used for disproving hypotheses.

Being skilled at finding counterexamples is a valuable skill in mathematical reasoning and problem solving.

Mathematical Notation

Mathematical notation is a standardized way of communicating complex mathematical ideas succinctly and precisely. It uses symbols and numbers to express operations, expressions, and equations.

• For instance, in our exercise, \(x \uparrow y\) was used to denote exponentiation, which can be more commonly represented as \(x^y\).
• Understanding these symbols is crucial for correctly interpreting and solving problems.

Sometimes symbols might differ across contexts, but they generally have widely accepted meanings. This standardization is critical because it allows mathematicians to communicate ideas clearly. Learning the meanings and applications of these notations is a fundamental step in advancing your mathematical proficiency.