1.1 Proof

• What is a mathematical proof?

  A logical argument
• Mathematical induction is used to prove statements for all natural numbers
• Mathematical induction can be used to prove formulas for all natural numbers
• Mathematical induction is used to prove statements for all natural numbers
• Proof by contradiction assumes the statement is false
• How would you start a direct proof to show that the sum of two even integers is even?

  Let x and y be even
• Proof by induction requires proving a base case and then showing that if it holds for n, it holds for n+1
• Steps in a proof by contradiction

  1️⃣ Assume the statement is false

  2️⃣ Derive a logical contradiction

  3️⃣ Conclude the original statement is true
• Proof by induction requires proving a base case and showing that if the statement holds for n, it also holds for n+1
• Direct proof starts with known facts and uses logical reasoning to derive new statements
• What number is commonly used to demonstrate proof by contradiction?

  √2
• Steps in a proof by induction

  1️⃣ Prove the base case

  2️⃣ Assume the statement is true for n

  3️⃣ Prove the statement is true for n+1
• In proof by induction, the induction step requires assuming the statement is true for n
• If a formula holds for n, it must also hold for n+1 in proof by induction


  True
• One type of proof method is called deduction, which starts with known facts
• Steps in mathematical induction

  1️⃣ Prove the base case

  2️⃣ Assume the statement holds for n

  3️⃣ Show it holds for n+1
• Proof by contradiction starts by assuming the statement is true

  False
• What is an example of a statement that can be proven using proof by contradiction?

  √2 is irrational
• What is the induction hypothesis in mathematical induction?

  Assume it holds for n
• The sum of two even integers can be written as 2a + 2b, where a and b are integers


  True
• A mathematical proof is a logical argument that demonstrates a statement is true based on axioms and previously proven theorems
• Match the proof method with its approach:

  Deduction ↔️ Logically progress from known facts

  Contradiction ↔️ Show that assuming the statement is false leads to a contradiction

  Induction ↔️ Prove a base case and show it holds for n+1 if it holds for n
• Proof by contradiction assumes the statement is true.

  False
• In a direct proof, each logical step must be clearly explained.


  True
• Proof by contradiction concludes that the original statement is true because assuming it was false led to a contradiction
• What is typically the base case used in proof by induction?

  n=1
• What is the induction step in proof by induction?

  Show n -> n+1
• What is the starting point of deduction in mathematical proof?

  Known facts
• Steps in proof by induction

  1️⃣ Show the base case (e.g., n=1)

  2️⃣ Assume the statement is true for n

  3️⃣ Prove the statement is true for n+1
• In proof by induction, if the formula holds for n, it must also hold for n+1


  True
• What is the approach in proof by deduction?

  Logically progress
• What is the use case for proof by contradiction?

  Proving irrationality
• What proof technique is used to show that √2 is irrational?

  Proof by Contradiction
• Deductive proof starts with known facts and uses logical reasoning to reach a conclusion.

  True
• What is assumed in proof by contradiction to begin the proof process?

  The statement is false
• Mathematical induction is used to prove statements true for all natural numbers.
• In Mathematical Induction, the formula for the sum of the first n integers is n(n+1)/2.
• What example is provided to illustrate a direct proof in the study material?

  Sum of two even integers
• What is the initial assumption when proving √2 is irrational using contradiction?

  √2 is rational
• Proof by contradiction assumes the statement is false