Problem 66 - Discrete Mathematics with Applications

Question

Give a counterexample to disprove each statement. $$x \downarrow(y \downarrow z)=(x \downarrow y) \downarrow z$$

Step-by-Step Solution

Answer

#Question# Provide a counterexample to disprove the following statement: "The \(\downarrow\) operation is associative, i.e., \(x \downarrow(y \downarrow z) = (x \downarrow y) \downarrow z\)." #Answer# Counterexample: Choose the subtraction operation for \(\downarrow\), and let x = 1, y = 2, and z = 3. Then, \(x \downarrow(y \downarrow z) = 1 - (2 - 3) = 2\), and \((x \downarrow y) \downarrow z = (1 - 2) - 3 = -4\). Since \(2 \neq -4\), the statement is false.

Step 1: Choose an operation

Since the exercise does not specify the operation \(\downarrow\), let’s consider a simple non-associative operation, namely subtraction. So, we have: $$ x - (y - z) \neq (x - y) - z $$

Step 2: Choose values for x, y, and z

Let's choose some values for x, y, and z to see if the statement is not true. Suppose we let x = 1, y = 2, and z = 3. Then the left-hand side of the equation is: $$ 1 - (2 - 3) = 1 - (-1) = 2 $$ And the right-hand side is: $$ (1 - 2) - 3 = (-1) - 3 = -4 $$

Step 3: Compare the results

Comparing the results of the left-hand side and right-hand side, we see that they are not equal: $$ 2 \neq -4 $$ Since we found a counterexample with x = 1, y = 2, and z = 3, we can conclude the original statement is false: $$ x \downarrow(y \downarrow z) \neq (x \downarrow y) \downarrow z $$

Key Concepts

Non-Associative Operation

In mathematics, an operation is said to be non-associative if it does not always yield the same result when grouping its operands in different ways. Simply put, switching the parenthesis or order of operations in non-associative operations changes the outcome. A simple example is subtraction, as shown in the original exercise. In subtraction, the order in which you subtract numbers matters.

For instance:

• Consider three numbers, 1, 2, and 3.
• If you subtract them as \( 1 - (2 - 3) \), you first do \( 2 - 3 = -1 \), and then \( 1 - (-1) = 2 \).
• But, if you subtract them as \( (1 - 2) - 3 \), it first becomes \( 1 - 2 = -1 \), and then \( -1 - 3 = -4 \).

Clearly, \( 2 eq -4 \), which shows that subtraction is non-associative as the grouping affects the final result. Understanding this property helps in identifying operations that can lead to different outcomes when the sequence is altered.

Logical Operators

Logical operators are symbols or words used to connect two or more expressions logically. They create compound statements, facilitate decision-making, and are key in fields like computer science and mathematics. Common logical operators include AND (\( \land \)), OR (\( \lor \)), and NOT (\( \lnot \)).

In the context of this exercise, the operation \( \downarrow \) isn't specified as a known logical operator. However, when dealing with logical operators, the focus is often on their properties, like associativity and distributivity, which relate to how they handle various true or false inputs.

For instance, logical OR (\( \lor \)) is associative:

• \( (A \lor B) \lor C \equiv A \lor (B \lor C) \)

This associative property means the order doesn't affect the outcome. If \( \downarrow \) were similar to logical OR, these properties would need to apply. Understanding logical operators and their nuances is crucial for analyzing logical statements and mathematical expressions.

Mathematical Proof

A mathematical proof is a logical argument that confirms the truth of a given statement. It utilizes axioms, theorems, and logical reasoning to arrive at a conclusion. Proofs are the foundational structure upon which mathematics builds reliable knowledge.

There are different forms of proof techniques:

• Direct proof: Directly shows that a statement is true based on established facts.
• Indirect proof or proof by contradiction: Assumes the opposite of the statement is true and derives a contradiction.
  This technique undermines the negated statement, thus proving the original statement true.
• Proof by induction: Useful for statements about integers, this technique proves the base case and then shows that if the statement holds for one integer, it holds for the next.

Proofs don't merely check if a statement seems true, they establish truth within a consistent framework of rules and definitions. The exercise under consideration may require you to invalidate (disprove) a statement through proof, specifically by counterexample.

Disproving Statements

Disproving a statement often involves finding a single counterexample that contradicts the statement's claim, demonstrating its falsehood. This is a vital part of mathematical reasoning, using evidence effectively to show that a statement does not hold under specific circumstances.

For example, in our exercise, if we assert "\( x \downarrow(y \downarrow z\) is the same as \( (x \downarrow y) \downarrow z \),” establishing their equality would require they yield the same results every time. However:

• By setting \( x = 1, y = 2, z = 3 \), the expression \( x - (y - z) \) results in \( 2 \).
• On the other hand, \( (x - y) - z \) results in \(-4 \).

Since \( 2 eq -4 \), the counterexample clearly disproves that these expressions are always equal. Disproving a statement requires patience and logical precision, often testing various cases until finding one that breaks the statement.