Mathematical Proof and Logic Rules Quiz

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Questions and Answers

What is the strategy used in proofs by contradiction?

Assume the hypothesis is true and the conclusion is false. (correct)

Assume the hypothesis and conclusion are true.

Negate both the hypothesis and conclusion.

Prove the hypothesis directly.

In the proof that n² is even implies n is even, what does assuming n is odd lead to?

A direct proof.

A counterexample.

A contradiction. (correct)

An implication.

If P → Q is proved by contradiction, what conclusion can be drawn?

Q → P is true.

P → Q is sometimes true.

P → Q is true. (correct)

P → Q is false.

In the proof that (3n+2) odd implies n is odd, why do we assume n is even?

To create a contradiction. Signup and view all the answers

What must be concluded if a proof by contradiction reaches a contradiction?

The assumption was false. Signup and view all the answers

Why do proofs by contradiction involve deriving a contradiction?

To show the assumption must be incorrect. Signup and view all the answers

Based on the provided text, when is an integer considered even?

When it is equal to 2k, where k is an integer. Signup and view all the answers

In the proof by contradiction example, what was the initial assumption made about the existence of a largest integer?

There is a largest integer. Signup and view all the answers

What method was used in the example to prove that there is no largest integer?

Proof by Contradiction Signup and view all the answers

In the contrapositive statement 'If not Q, then not P', what does 'not P' represent in the original statement 'If P, then Q'?

The hypothesis Signup and view all the answers

How did the author conclude that an integer must be odd?

By contradicting the assumption that the integer was even. Signup and view all the answers

In notation, how is 'not P' represented?

~P Signup and view all the answers

What is the definition of an even integer according to the text?

An integer n is even if there is some integer k such that n = 2k. Signup and view all the answers

Which of the following statements represents an implication according to the text?

If P, then Q. Signup and view all the answers

What is a direct proof according to the text?

A proof that starts with an initial set of hypotheses and proves the conclusion using simple logical steps. Signup and view all the answers

Which statement correctly describes integer relationships based on the text?

No integer is both even and odd. Signup and view all the answers

What should be the characteristics of the axioms mentioned in the text?

They should be minimal for simplicity. Signup and view all the answers

Which type of logical reasoning involves proving a statement by assuming its negation and reaching a contradiction?

Proof by Contradiction Signup and view all the answers

Study Notes

Proofs by Contradiction

In proofs by contradiction, a statement's negation is assumed to be true, and then this assumption is shown to lead to a logical contradiction, thereby proving the original statement.
The strategy involves deriving a contradiction to conclude the truth of the original statement.

Examples of Proofs by Contradiction

In the proof that n² is even implies n is even, assuming n is odd leads to a contradiction, thereby proving that n must be even.
In the proof that (3n+2) odd implies n is odd, n is assumed to be even, and then a contradiction is derived to conclude that n must be odd.

Implications and Contrapositive Statements

If P → Q is proved by contradiction, it can be concluded that if P is true, then Q must also be true.
In a contrapositive statement 'If not Q, then not P', 'not P' represents the negation of the original statement 'If P, then Q'.
'Not P' is represented in notation as ~P.

Properties of Integers

An integer is considered even if it can be written in the form 2k, where k is an integer.
The definition of an even integer is an integer that can be divided by 2 without leaving a remainder.

Direct Proofs and Axioms

A direct proof involves showing that a statement is true through a series of logical steps, without assuming its negation.
Axioms are statements that are assumed to be true without needing to be proven, and should be self-evident, consistent, and non-redundant.

Logical Reasoning

Proving a statement by assuming its negation and reaching a contradiction is a type of logical reasoning called proof by contradiction.
In this type of reasoning, the goal is to show that the negation of the statement leads to a logical contradiction, thereby proving the original statement.

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Description

This quiz explores the concept of mathematical proofs, including how they connect axioms, definitions, and validated statements to demonstrate the correctness of a statement. It emphasizes the importance of logic rules and agreed-upon axioms, along with the appropriate level of detail for different audiences. Reference book: Thomas Cormen's 'Structure of a Mathematical Proof'.