18 Questions
What is the strategy used in proofs by contradiction?
Assume the hypothesis is true and the conclusion is false.
In the proof that n² is even implies n is even, what does assuming n is odd lead to?
A contradiction.
If P → Q is proved by contradiction, what conclusion can be drawn?
P → Q is true.
In the proof that (3n+2) odd implies n is odd, why do we assume n is even?
To create a contradiction.
What must be concluded if a proof by contradiction reaches a contradiction?
The assumption was false.
Why do proofs by contradiction involve deriving a contradiction?
To show the assumption must be incorrect.
Based on the provided text, when is an integer considered even?
When it is equal to 2k, where k is an integer.
In the proof by contradiction example, what was the initial assumption made about the existence of a largest integer?
There is a largest integer.
What method was used in the example to prove that there is no largest integer?
Proof by Contradiction
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
How did the author conclude that an integer must be odd?
By contradicting the assumption that the integer was even.
In notation, how is 'not P' represented?
~P
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.
Which of the following statements represents an implication according to the text?
If P, then Q.
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.
Which statement correctly describes integer relationships based on the text?
No integer is both even and odd.
What should be the characteristics of the axioms mentioned in the text?
They should be minimal for simplicity.
Which type of logical reasoning involves proving a statement by assuming its negation and reaching a contradiction?
Proof by Contradiction
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'.
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