Introduction to Formal Proof Techniques

Question 1

What is a direct proof?

Answer 1

A direct proof establishes the truth of a statement by a straightforward application of axioms, definitions, and previously established theorems.

Question 2

In what cases is direct proof most effectively used?

Answer 2

Direct proof is most effective when a straightforward logical path exists from the premises to the desired conclusion.

Question 3

What is proof by contradiction?

Answer 3

Proof by contradiction establishes the truth of a statement by showing that assuming its negation leads to a contradiction.

Question 4

Why is proof by contradiction a powerful technique?

Answer 4

It’s powerful because it can resolve complex proofs by demonstrating that the opposite scenario is impossible.

Question 5

What is the starting point in a proof by contradiction?

Answer 5

The starting point is the assumption that the statement to be proved is false.

Question 6

What is proof by contraposition?

Answer 6

Proof by contraposition proves a statement “If P, then Q” by proving its contrapositive “If not Q, then not P.”

Question 7

How is proof by contraposition logically equivalent to direct proof?

Answer 7

Both prove the same statement, but proof by contraposition approaches it by establishing the truth of its contrapositive.

Question 8

Can all statements be proven directly?

Answer 8

Not all statements can be proven directly; some require indirect methods like proof by contradiction or contraposition.

Question 9

Give an example where direct proof is typically used.

Answer 9

Direct proof is used in proving statements like “If n is an even integer, then n² is even.”

Question 10

Provide an example where proof by contradiction is suitable.

Answer 10

Proof by contradiction is suitable for statements like the irrationality of the square root of 2.

Question 11

How does one conclude a proof by contradiction?

Answer 11

The proof concludes by demonstrating that the assumption leads to an inconsistency, thereby affirming the original statement.

Question 12

What is an essential aspect of setting up a proof by contraposition?

Answer 12

Identifying the correct contrapositive statement is crucial in setting up the proof effectively.

Question 13

In proof by contradiction, what happens if no contradiction is reached?

Answer 13

If no contradiction is reached, the proof is unsuccessful, and a different proof strategy might be required.

Question 14

How do you demonstrate a statement is false using direct proof?

Answer 14

Direct proof isn’t typically used to prove a statement false; it’s used to prove implications or positive assertions.

Question 15

Why might one choose proof by contraposition over direct proof?

Answer 15

Proof by contraposition might be chosen when the contrapositive statement is simpler or more direct to prove than the original.

Question 16

Is proof by contradiction accepted in all mathematical fields?

Answer 16

While widely accepted, some areas, particularly constructive mathematics, do not accept proof by contradiction.

Question 17

What role do definitions play in direct proofs?

Answer 17

Definitions provide the foundational meanings of terms and concepts, which are critical for logical progression in direct proofs.

Question 18

Can proof by contraposition be used to prove existence statements?

Answer 18

No, it’s generally not used for existence statements but rather for implications or conditional statements.

Question 19

How does one verify the success of a proof by contradiction?

Answer 19

The success is verified when the assumption logically leads to an untenable or contradictory conclusion.

Question 20

What distinguishes direct proof from other proof techniques?

Answer 20

Direct proof follows a linear logical progression from premises to conclusion without assuming the negation or considering alternative cases.