6.1 Methods of Proof

• What are the key methods used in mathematical proofs?

  Direct, contradiction, induction
• Direct Proof is a method where you start with the given assumptions and use logic
• Proof by Contradiction begins by assuming the negation of the statement is true.
• What is the contrapositive of the statement $P \implies Q$?


  $\neg Q \implies \neg P$
• Direct Proof uses given assumptions to arrive at the desired conclusion
• Indirect Proof includes Proof by Contradiction and Proof by Contrapositive.
• Match the type of proof with its approach:

  Proof by Contradiction ↔️ Assumes negation of conclusion

  Proof by Contrapositive ↔️ Proves the contrapositive
• Steps involved in a Direct Proof

  1️⃣ Start with given assumptions

  2️⃣ Apply logical rules and theorems

  3️⃣ Arrive at the conclusion
• What is the main difference between Proof by Contradiction and Proof by Contrapositive?

  Contradiction shows fallacy
• Indirect Proof starts by assuming the opposite of what needs to be proven.
• What are the two possible outcomes when using indirect proof methods?

  Contradiction or contrapositive
• In Proof by Contradiction, we start by assuming the negation of the conclusion
• If the negation of the conclusion in Proof by Contradiction leads to a contradiction, the original statement must be true.
• What is an example statement proved by contradiction in the material?

  $\sqrt{2}$ is irrational
• Steps in proving that $\sqrt{2}$ is irrational using Proof by Contradiction


  1️⃣ Assume $\sqrt{2}$ is rational

  2️⃣ $\sqrt{2} =$$\frac{a}{b}$ where $a$ and $b$ are coprime

  3️⃣ Show that $a^{2}$ is even

  4️⃣ Deduce that $a$ is even

  5️⃣ Substitute $a =$$2k$

  6️⃣ Show that $b^{2}$ is even

  7️⃣ Conclude $\sqrt{2}$ is irrational
• The contrapositive of $P \implies Q$ is \neg Q \implies \neg P
• In Proof by Contrapositive, if $n$ is odd, then $n^{2}$ is odd.
• What is the primary difference between Proof by Contradiction and Proof by Contrapositive?

  Assumed to be true
• Indirect Proof starts by assuming the opposite of what we want to prove
• The contrapositive of "If $n^{2}$ is even, then $n$ is even" is "If $n$ is odd, then $n^{2}$ is odd."
• Steps in Proof by Contradiction

  1️⃣ State the proposition $P$ to be proved

  2️⃣ Assume the negation of $P$, i.e., $\neg P$, is true

  3️⃣ Use logical deduction to derive a contradiction

  4️⃣ Show that this contradiction proves $P$ is true

  5️⃣ Conclude the proof
• What is an example statement proved by contradiction in the material?

  $\sqrt{2}$ is irrational
• Proof by Induction is used to prove statements that hold for all natural numbers
• Steps in Proof by Induction

  1️⃣ Base Case: Show the statement is true for $n =$$1$

  2️⃣ Inductive Hypothesis: Assume the statement is true for $n =$$k$

  3️⃣ Inductive Step: Prove the statement is true for $n =$$k +$$1$ using the inductive hypothesis
• What is an example statement proved by induction in the material?

  $\sum_{i = 1}^{n} i =$$\frac{n(n + 1)}{2}$
• The scope of Proof by Induction is statements about all natural numbers
• Match the proof method with its description:

  Direct Proof ↔️ Starts with assumptions and uses logic

  Proof by Contradiction ↔️ Assumes the negation of the conclusion

  Proof by Induction ↔️ Proves statements for all natural numbers
• What is the starting point of a direct proof?

  Given assumptions
• If $n$ is an even integer, then $n^{2}$ is a multiple of 2
• Proof by contradiction begins by assuming the negation of the conclusion.
• What number is commonly used in proof by contradiction to demonstrate irrationality?

  $\sqrt{2}$
• Steps in proving $\sqrt{2}$ is irrational using proof by contradiction


  1️⃣ Assume \sqrt{2}</latex> is rational, i.e., $\sqrt{2} =$$\frac{a}{b}$

  2️⃣ Derive $a^{2} =$$2b^{2}$

  3️⃣ Show $a$ is even, so $a =$$2k$

  4️⃣ Show $b$ is also even

  5️⃣ Conclude $\sqrt{2}$ is irrational
• The contrapositive of $P \implies Q$ is $\neg Q \implies \neg P$.
• If $n$ is odd, then $n^{2}$ is also odd
• What is the key approach in direct proof?

  Logical deduction
• Proof by contradiction starts by assuming the negation of the conclusion
• The contrapositive of $P \implies Q$ is $\neg Q \implies \neg P$.
• Steps in a proof by contradiction

  1️⃣ State the proposition $P$ to be proved

  2️⃣ Assume the negation of $P$, i.e., $\neg P$, is true

  3️⃣ Derive a logical contradiction

  4️⃣ Show that the contradiction proves $P$ is true

  5️⃣ Conclude the proof
• What is the assumption in a proof by contradiction?

  Negation of conclusion
• In proof by induction, the base case involves showing the statement is true for n = 1