3.2 A: Proof

• A mathematical proof is a logical argument that establishes the truth of a statement based on defined axioms
• A proof by contradiction assumes the statement is false
• Inductive proofs require verifying a base case to start.

  True
• What is the first step in constructing a direct proof?

  Identify assumptions
• What is a direct proof in mathematics?

  A logical sequence of steps
• What is a mathematical proof?

  A logical argument for truth
• Match the type of proof with its description:

  Direct Proof ↔️ Derives conclusion directly

  Proof by Contradiction ↔️ Assumes statement is false

  Inductive Proof ↔️ Proves for all natural numbers
• Proving that √2 is irrational is an example of proof by contradiction.

  True
• Steps to construct a proof by contradiction:

  1️⃣ State the assumption

  2️⃣ Derive a contradiction

  3️⃣ Conclude the original statement is true
• In a proof by contradiction, if the negation of a statement leads to a contradiction, the original statement must be true
• What conclusion is reached when using proof by contradiction to show that √2 is irrational?

  √2 is irrational
• Steps in an inductive proof

  1️⃣ Verify the base case

  2️⃣ Establish the inductive step
• Which proof strategy is used to show that √2 is irrational by assuming it is rational and deriving a contradiction?

  Proof by contradiction
• In the inductive step, it is assumed that the statement is true for some natural number n.
• The direct proof of the sum of the interior angles of a triangle relies on alternate interior angles.

  True
• The inductive step in mathematical induction involves assuming the statement is true for some natural number n.


  True
• A mathematical proof is a logical argument based on defined axioms, definitions, and previously proven results.
• Mathematical induction requires verifying both a base case and an inductive step.


  True
• In a proof by contradiction, the first step is to assume the statement is false.
• The conclusion of a proof is the final proven statement.


  True
• In a proof by contradiction, deriving a contradiction means the original statement must be true.

  True
• In a proof by contradiction, you derive a contradiction from the assumption that the statement is false.
• The inductive step shows that if the statement is true for n, it is also true for n+1.

  True
• Inductive proofs are used to prove statements for all natural numbers.


  True
• A proof by contradiction assumes the negation of the statement and shows this leads to a contradiction
• An inductive proof begins by verifying the statement for a base case
• What is an example of a theorem proven using direct proof?

  Congruence theorems
• Steps to construct a direct proof

  1️⃣ Identify the assumptions and conclusion

  2️⃣ Use logical reasoning

  3️⃣ State the proven conclusion
• A proof by contradiction concludes that the original statement is true because its negation leads to a contradiction.
• What two steps are required to complete an inductive proof?

  Verify base case and inductive step
• What is the formula for the sum of the first n natural numbers that is proven using induction?

  $\frac{n(n + 1)}{2}$
• Different proof strategies are best understood through examples.


  True
• The proof by contradiction is used to show that $\sqrt{3}$ is irrational.


  True
• In an inductive proof, the starting point is called the base
• Proof by contradiction requires showing that the assumption leads to a logical contradiction.

  True
• What is the key idea in a proof by contradiction?

  Derive a contradiction
• Why is $2(a + b)$ a multiple of 2 in the direct proof example?


  Because (a + b) is an integer
• What conclusion is derived when $\sqrt{3}$ is assumed to be rational?


  Both p and q are divisible by 3
• Deduction in proofs moves from general statements to specific conclusions.

  True
• Match the proof type with its definition:

  Deduction ↔️ Moves from general to specific cases

  Induction ↔️ Proves statements for natural numbers

  Contradiction ↔️ Assumes the statement is false