Problem 3 - Analysis with an introduction to proof

Question

Write the negation of each statement. it (a) The \(3 \times 3\) identity matrix is singular. (b) The function \(f(x)=\sin x\) is bounded on \(\mathbb{R}\). (c) The functions \(f\) and \(g\) are linear. (d) Six is prime or seven is odd. (e) If \(x\) is in \(D\), then \(f(x)<5\). (f) If \(\left(a_{n}\right)\) is monotone and bounded, then \(\left(a_{n}\right)\) is convergent. (g) If \(f\) is injective, then \(S\) is finite or denumerable.

Step-by-Step Solution

Answer

(a) The \(3 \times 3\) identity matrix is not singular. (b) The function \(f(x)=\sin x\) is not bounded on \(\mathbb{R}\). (c) The functions \(f\) and \(g\) are not both linear. (d) Six is not prime and seven is not odd. (e) There exists an \(x\) in \(D\) such that \(f(x) \geq 5\). (f) There exists a sequence \(\left(a_{n}\right)\) which is monotone and bounded but not convergent. (g) There exists an injective function \(f\) such that \(S\) is neither finite nor denumerable.

Step 1: (a) Negation of the statement about the identity matrix

: The original statement is: "The \(3 \times 3\) identity matrix is singular." To negate the statement, we should say the opposite of the original statement: "The \(3 \times 3\) identity matrix is not singular."

Step 2: (b) Negation of the statement about the function \(f(x) = \sin x\)

: The original statement is: "The function \(f(x)=\sin x\) is bounded on \(\mathbb{R}\)." To negate the statement, we should say the opposite of the original statement: "The function \(f(x)=\sin x\) is not bounded on \(\mathbb{R}\)."

Step 3: (c) Negation of the statement about functions \(f\) and \(g\)

: The original statement is: "The functions \(f\) and \(g\) are linear." To negate the statement, we should say the opposite of the original statement: "The functions \(f\) and \(g\) are not both linear." (At least one of them is not linear)

Step 4: (d) Negation of the statement about six and seven

: The original statement is: "Six is prime or seven is odd." To negate the statement, we should say the opposite of the original statement: "Six is not prime and seven is not odd."

Step 5: (e) Negation of the conditional statement about \(x\) and \(f(x)\)

: The original statement is: "If \(x\) is in \(D\), then \(f(x) < 5\)." To negate the statement, we should say the opposite of the original statement: "There exists an \(x\) in \(D\) such that \(f(x) \geq 5\)."

Step 6: (f) Negation of the conditional statement about sequences

: The original statement is: "If \(\left(a_{n}\right)\) is monotone and bounded, then \(\left(a_{n}\right)\) is convergent." To negate the statement, we should say the opposite of the original statement: "There exists a sequence \(\left(a_{n}\right)\) which is monotone and bounded but not convergent."

Step 7: (g) Negation of the conditional statement about injective functions

: The original statement is: "If \(f\) is injective, then \(S\) is finite or denumerable." To negate the statement, we should say the opposite of the original statement: "There exists an injective function \(f\) such that \(S\) is neither finite nor denumerable."

Key Concepts

Negation of Statements

In mathematical logic, negation is the process of reversing the truth value of a statement. When we negate a statement, we aim to express the opposite of its original meaning. For example, if the statement is true, its negation will be false, and vice versa. Negation is a fundamental concept needed to clearly express disagreements, and inconsistencies, or to simply analyze mathematical propositions.

Let's consider the statement, "The function \( f(x)=\sin x \) is bounded on \( \mathbb{R} \)." Bounded means there exists a limit beyond which the function does not go. Therefore, to negate it, you say that "The function \( f(x)=\sin x \) is not bounded on \( \mathbb{R} \)," implying there is no limit.

Another aspect involves compound statements. For instance, consider "Six is prime or seven is odd." Negating this means that both conditions must be false: "Six is not prime and seven is not odd." It is crucial to change "or" to "and," showing a distinction in how conjunctions and disjunctions are handled in negation.

• The negation of "and" is "or" (and vice versa).
• Use quantifiers for global and existential properties.
• Understand the logical operators well for successful negation.

Conditional Statements

Conditional statements, or "if-then" statements, play a crucial role in mathematical reasoning. These are written in the form "If \( P \), then \( Q \)," where \( P \) is called the hypothesis and \( Q \) the conclusion. The truth of a conditional statement hinges on the logical relationship between these two parts.

Consider the statement "If \( x \) is in \( D \), then \( f(x) < 5 \)." Here, our task is to negate the conditional statement. The negation involves showing a scenario where the hypothesis holds, but the conclusion does not. Thus, its negation would be "There exists an \( x \) in \( D \) such that \( f(x) \geq 5 \)."

For effective negation of such statements, remember:

• Not every instance of \( P \) leads to \( Q \).
• Identify existence vs. universal quantification.
• Switch from "for all" to "there exists" (and vice versa) appropriately in negations.

Constructing and negating conditional statements strengthens logical deduction skills and enhances mathematical problem-solving capabilities.

Mathematical Proof

Mathematical proof is a logical argument that verifies the truth of a statement using accepted truths, definitions, and inferential reasoning. Proofs are the bedrock of mathematics, used to establish undeniable truths.

Several techniques are employed in crafting a proof, including direct proof, proof by contradiction, and proof by contrapositive. Each provides a unique way to uncover truth based on the original statement and its logical structure.

A proof by contrapositive can be incredibly useful, especially with conditional statements. Instead of proving "If \( P \), then \( Q \)," you prove "If not \( Q \), then not \( P \)." This is because these two are logically equivalent.

• Understand different proof techniques.
• Develop strategies based on statement types.
• Learn to identify premises and conclusions.

Exploring these proof strategies provides robust skills in not only solving mathematical problems but also in constructing strong logical arguments. In all, mathematical proofs teach precise thinking and articulate reasoning.