Question

A fallacy Explain the fallacy in the following argument. Let \(x=1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots\) and \(y=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\cdots \cdot\) It follows that \(2 y=x+y\) which implies that \(x=y .\) On the other hand, $$ x-y=\underbrace{\left(1-\frac{1}{2}\right)}_{>0}+\underbrace{\left(\frac{1}{3}-\frac{1}{4}\right)}_{>0}+\underbrace{\left(\frac{1}{5}-\frac{1}{6}\right)}_{>0}+\cdots>0 $$ is a sum of positive terms, so \(x>y .\) Therefore, we have shown that \(x=y\) and \(x>y\)

Short answer

Question: Identify and explain the fallacy in the given argument that concludes both x=y and x>y. Answer: The fallacy in the given argument arises from the incorrect combination of equations and inequalities, leading to the mistaken claims of both x=y and x>y. The true relationship between x and y is established by the equation x=y, and the fallacy occurs when introducing inequalities (order comparison) in the argument, trying to force two conflicting conclusions. It is important to examine additional conclusions based on inequalities critically to ensure logical consistency in the argument.

Step-by-step solution

Identifying the contradiction

The given argument demonstrates a contradiction by claiming that \(x=y\) and \(x>y\) at the same time. A contradiction means there is an error in the argument somewhere, and our task is to find it.

Examining the argument

Looking at the argument, we see that it consists of several steps: 1. It represents two infinite series \(x\) and \(y\) as given. 2. Claims that \(2y=x+y\). 3. States that this implies \(x=y\). 4. From the difference of the given series \(x-y\), it concludes that \(x>y\). There must be a mistake in either the premises or the logic of these steps for the contradiction to occur.

Evaluating the logic

We notice that the second step claims that \(2y=x+y\), which implies \(x=y\). This is a correct implication: if \(2y=x+y\) is true, then indeed \(x=y\). However, the fourth step, where it is said that \(x>y\) due to the difference of the series summing to positive terms \((x-y)>0\), is dissimilar to the steps before it. Contrary to earlier steps where we had algebraic identities, now we have inequalities in the argument. Inequalities are not equivalent to equations - they express a relationship of ordering between the elements in question, whereas equations express the equality of the elements. Combining equations and inequalities together in this manner is where the error lies in the argument.

Identifying the fallacy

The fallacy in the argument is found in the step where \(x>y\) is claimed based on the difference of the series sums \((x-y)>0\). This step incorrectly assumes that the relationship of \(x>y\) is justified by this inequality – but the true relationship between the sums was already established when \(x=y\) was found in a previous step. By mixing the context of an equation (equality) with an inequality (order comparison), the argument tries to force two conflicting conclusions, which is the reason for the contradiction.

Conclusion

The fallacy in the given argument arises from the incorrect combination of equations and inequalities, leading to the mistaken claims of both \(x=y\) and \(x>y\). The true relationship between \(x\) and \(y\) is established by the equation \(x=y\), and any additional conclusions based on inequalities should be critically examined to ensure the logical consistency of the argument.

Key concepts

Infinite Series

An infinite series is a sum of an infinite sequence of numbers. It is often represented in the form of \(a_1 + a_2 + a_3 + \cdots\). The series continues indefinitely, which means we keep adding terms without any end.
The example given here mentions two series, \(x = 1 + \frac{1}{3} + \frac{1}{5} + \cdots\) and \(y = \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \cdots\). Both are infinite series, suggesting that they
represent ongoing sums of fractions.

• In mathematics, some infinite series converge, meaning they approach a specific finite value. Others diverge, continuing to grow larger indefinitely.
• The terms in an infinite series can be positive, negative, or alternating. This impacts whether they converge or diverge.
• Understanding infinite series is crucial when solving problems involving summation and sequence analysis.

In the original exercise, recognizing these sequences as infinite series immediately gives insight into why a comparison between them using algebraic operations like equations and inequalities might lead to unexpected results.

Equations vs Inequalities

Equations and inequalities are two fundamental concepts used to express relationships in mathematics. They have distinct meanings and uses.

• Equations: These involve equals signs to show that two expressions are identical in value. For example, \(2 + 3 = 5\) indicates both sides have the same value.
• Inequalities: These express a relationship of order. They show that one side is greater than, less than, or not equal to the other. For instance,\(4 > 2\) (4 is greater than 2).

The original argument mixes these two by attempting to apply both equations and inequalities simultaneously, leading to confusion.


The claim \(x = y\) comes from an equation, suggesting equality of infinite series sums. However, the inequality assertion \(x > y\) comes from observing that each individual term in the series differences \(x - y\) contributes positively, implying one series is inherently larger. This contradiction arises because we can't juxtapose equality with order without understanding their implications, which in turn manifests a mathematical fallacy within the logic.

Contradiction in Mathematical Arguments

Contradiction is a situation where we arrive at two supposedly valid conclusions that contradict each other.This occurs when the logic or premises employed in an argument have inherent flaws.

In the exercise, the contradiction presents itself by claiming both \(x=y\) and \(x>y\). This signalizes a fallacy has occurred,usually indicating a misstep or oversight in an earlier logical process or assumptions.

• Identifying contradictions involves closely examining each step of an argument for consistency.
• You should check the initial assumptions, calculations, and logical deductions for errors or deviations.
• Concluding with mutually exclusive results like equalities and inequalities often points towards incorrect reasoning or misapplied principles.

The exercise demonstrates how contradictions arise when an argument improperly mixes equality with ordering without supporting logical backing. Recognizing these contradictions helps us learn from and correct our logical approach, ensuring we arrive at valid solutions.