Proofs Flashcards
(24 cards)
What is a direct proof?
A direct proof is a method of proving a statement by straightforwardly demonstrating that the conclusion follows logically from the premises.
True or False: In a direct proof, you can use assumptions that are not relevant to the conclusion.
False
What does proof by contraposition involve?
Proof by contraposition involves proving that if the conclusion is false, then the hypothesis must also be false.
Fill in the blank: In a proof by contradiction, you assume that the conclusion is _____ and show that this leads to a contradiction.
false
What is the main goal of proof by cases?
The main goal of proof by cases is to prove a statement by dividing it into several cases and proving each case separately.
True or False: Proof by cases can only be used when there are a finite number of cases to consider.
True
What is the structure of a proof by contradiction?
The structure involves assuming the negation of the statement to be proved and deriving a contradiction from this assumption.
Multiple Choice: Which of the following is NOT a common method of proof? A) Direct Proof B) Proof by Contradiction C) Proof by Induction D) Proof by Assumption
D) Proof by Assumption
Define proof by contraposition in one sentence.
Proof by contraposition is a method where the implication ‘if P then Q’ is proven by demonstrating ‘if not Q then not P’.
True or False: In a direct proof, you can use examples to establish the truth of a general statement.
False
Fill in the blank: A proof that demonstrates an implication by showing that the contrapositive is true is called a _____ proof.
contraposition
What is the first step in a proof by contradiction?
The first step is to assume that the statement you want to prove is false.
Multiple Choice: Which proof method is most useful when dealing with an ‘or’ statement? A) Direct Proof B) Proof by Cases C) Proof by Contradiction D) Proof by Contraposition
B) Proof by Cases
True or False: Proof by cases requires that each case considered must cover all possible scenarios.
True
What is the key difference between direct proof and proof by contradiction?
The key difference is that direct proof establishes the truth of a statement directly, while proof by contradiction starts by assuming the statement is false.
Fill in the blank: In proofs, the term ‘contradiction’ refers to a situation where two or more _____ statements cannot be true at the same time.
logical
What is an example of a statement that can be proved by contraposition?
If it is raining, then the ground is wet can be proved by showing that if the ground is not wet, then it is not raining.
True or False: Proof by cases can only be applied to finite sets of conditions.
False
What is the conclusion of a proof by contradiction?
The conclusion is that the original assumption (the statement being false) must be incorrect, thus proving the statement true.
Fill in the blank: A proof that relies on an exhaustive list of cases is known as a _____ proof.
proof by cases
What is the primary purpose of a proof in mathematics?
The primary purpose of a proof is to demonstrate the truth of a statement or theorem based on axioms and previously established results.
Multiple Choice: Which method is typically used to prove theorems involving universal quantifiers? A) Direct Proof B) Proof by Contradiction C) Proof by Cases D) All of the above
D) All of the above
Short Answer: What is the relationship between direct proof and the law of non-contradiction?
Direct proof relies on the law of non-contradiction, which states that contradictory statements cannot both be true at the same time.
True or False: Proof by contradiction can sometimes be more difficult to construct than direct proofs.
True
