Problem 8 - A Gentle Introduction to the Art of Mathematics

Question

Find the contrapositives of the following sentences. (a) If you can't do the time, don't do the crime. (b) If you do well in school, you'll get a good job. (c) If you wish others to treat you in a certain way, you must treat others in that fashion. (d) If it's raining, there must be clouds. (e) If \(a_{n} \leq b_{n},\) for all \(n\) and \(\sum_{n=0}^{\infty} b_{n}\) is a convergent series, then \(\sum_{n=0}^{\infty} a_{n}\) is a convergent series.

Step-by-Step Solution

Answer

(a) If you do the crime, you must be able to do the time. (b) If you don't get a good job, you didn't do well in school. (c) If you don't treat others in that fashion, you don't wish others to treat you in that way. (d) If there are no clouds, it's not raining. (e) If \(\sum_{n=0}^{\infty} a_n\) is not a convergent series, then \(a_n \leq b_n\) for all \(n\) and \(\sum_{n=0}^{\infty} b_n\) is not a convergent series.

Step 1: Understand the Concept of Contraposition

The contrapositive of an implication statement 'If P, then Q' is 'If not Q, then not P.'

Step 2: Identify P and Q for Sentence (a)

The sentence 'If you can't do the time, don't do the crime' can be broken down into: P = 'you can't do the time' and Q = 'don't do the crime.'

Step 3: Find the Contrapositive for Sentence (a)

The contrapositive is 'If you do the crime, you must be able to do the time.'

Step 4: Identify P and Q for Sentence (b)

The sentence 'If you do well in school, you'll get a good job' can be broken down into: P = 'you do well in school' and Q = 'you'll get a good job.'

Step 5: Find the Contrapositive for Sentence (b)

The contrapositive is 'If you don't get a good job, you didn't do well in school.'

Step 6: Identify P and Q for Sentence (c)

The sentence 'If you wish others to treat you in a certain way, you must treat others in that fashion' can be broken down into: P = 'you wish others to treat you in a certain way' and Q = 'you must treat others in that fashion.'

Step 7: Find the Contrapositive for Sentence (c)

The contrapositive is 'If you don't treat others in that fashion, you don't wish others to treat you in that way.'

Step 8: Identify P and Q for Sentence (d)

The sentence 'If it's raining, there must be clouds' can be broken down into: P = 'it's raining' and Q = 'there must be clouds.'

Step 9: Find the Contrapositive for Sentence (d)

The contrapositive is 'If there are no clouds, it's not raining.'

Step 10: Identify P and Q for Sentence (e)

The sentence 'If \(a_n \leq b_n\) for all \(n\) and \(\sum_{n=0}^{\infty} b_n\) is a convergent series, then \(\sum_{n=0}^{\infty} a_n\) is a convergent series' can be broken down into: P = '\(a_n \leq b_n\) for all \(n\) and \(\sum_{n=0}^{\infty} b_n\) is a convergent series' and Q = '\(\sum_{n=0}^{\infty} a_n\) is a convergent series.'

Step 11: Find the Contrapositive for Sentence (e)

The contrapositive is 'If \(\sum_{n=0}^{\infty} a_n\) is not a convergent series, then \(a_n \leq b_n\) for all \(n\) and \(\sum_{n=0}^{\infty} b_n\) is not a convergent series.'

Key Concepts

Logical Implications

A logical implication takes the form 'If P, then Q.' It means that whenever P is true, Q must also be true. In logic, P is called the 'antecedent,' and Q is the 'consequent.' Understanding logical implications is the foundation for many mathematical proofs and reasoning.

Let's break this down with an example from the exercise:
If you do well in school, you'll get a good job.
Here, the implication is:
P = 'You do well in school' (antecedent)
Q = 'You'll get a good job' (consequent).
The truth of Q hinges on the truth of P. If P happens, then Q is guaranteed to happen.

Proof Techniques

Proof techniques are methods used to establish the truth of mathematical statements. One common method is a direct proof, where you start by assuming P (the hypothesis) and logically deduce Q (the conclusion).

Another vital technique is **proof by contrapositive**. To prove 'If P, then Q,' it is equivalent to prove 'If not Q, then not P.'

For example, to prove 'If it's raining, there must be clouds' (P -> Q), we can prove its contrapositive 'If there are no clouds, it's not raining' (¬Q -> ¬P). If we can show that whenever there are no clouds it can't rain, this indirectly but effectively proves our original statement.

Using contrapositives can simplify proofs, especially when proving the direct statement is complex.

Convergent Series

A series is a sum of terms of a sequence. A series \(\text{ot Q, then not P\rightarrow∑_{n=0}^{∞}\) is **convergent** if its partial sums tend to a finite limit as \(n\) \(\rightarrow\) \(∞\).

For example: \(\rightarrow\) \(\text{\text{not P, then not Q) is a statement that states that not Q necessarily leads to not P.\rightarrowa_n\) be a sequence. The ...//text.

Consider the series summed not Q summation: \rightarrow$$.

If you wish to know about how not Q implies \rightarrow$.

Analyzing the convergence of a series is fundamental to understanding the behavior of infinite sums.