1.1 Proof

• A mathematical proof is based on axioms and previously established theorems.


  True
• Steps to prove that the sum of two even numbers is even.

  1️⃣ Let $x$ and $y$ be even numbers

  2️⃣ By definition, $x =$$2a$ and $y =$$2b$

  3️⃣ Then, $x +$$y =$$2a +$$2b =$$2(a + b)$

  4️⃣ Since $a +$$b$ is an integer, $x +$$y$ is a multiple of 2

  5️⃣ Therefore, $x +$$y$ is even
• What type of reasoning is used in a direct proof to arrive at the desired result?

  Step-by-step logical reasoning
• An indirect proof demonstrates the truth of a statement by showing that its negation leads to a logical contradiction
• Indirect proof is useful when a direct proof is difficult or not obvious.
• Steps in a proof by contradiction:

  1️⃣ Assume the negation of the statement

  2️⃣ Derive a logical contradiction

  3️⃣ Conclude the original statement is true
• In proof by contradiction, the negation of the statement leads to a contradiction, so the original statement must be true.
• What are the three main steps in a proof by induction?

  Base case, hypothesis, step
• Steps in a proof by induction:

  1️⃣ Base Case

  2️⃣ Inductive Hypothesis

  3️⃣ Inductive Step
• Logical reasoning is essential for constructing a valid mathematical proof.

  True
• Match the key element of a mathematical proof with its definition:

  Axioms ↔️ Fundamental statements accepted as true without proof.

  Theorems ↔️ Statements proven true using axioms and other theorems.

  Logical Reasoning ↔️ Application of logical principles to construct the proof.
• An indirect proof demonstrates the truth of a statement by showing that the negation of the statement leads to a contradiction.


  True
• $\sqrt{2}$ is an irrational number.


  True
• Steps in a proof by induction:

  1️⃣ Base Case: Verify the statement 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.
• Steps in a proof by induction

  1️⃣ Base Case: Show the statement is true for the smallest natural number, usually n = 1</latex>.

  2️⃣ Inductive Hypothesis: Assume the statement is true for some arbitrary natural number $k$, i.e., $P(k)$ is true.

  3️⃣ Inductive Step: Prove the statement is true for the next natural number $k +$$1$, i.e., show $P(k + 1)$ is true based on the assumption $P(k)$.
• In a direct proof, the conclusion is logically deduced from the given premises.
• Proof by Induction is used to prove statements for all natural numbers $n$.


  True
• Steps in proving by induction that the sum of the first $n$ natural numbers is $\frac{n(n + 1)}{2}$


  1️⃣ Base Case: Show it's true for $n =$$1$.

  2️⃣ Inductive Hypothesis: Assume it's true for some $k$.

  3️⃣ Inductive Step: Prove it's true for $k +$$1$ based on $P(k)$.
• A direct proof begins with given premises
• For which natural number is the base case usually tested in induction?

  $n =$$1$
• Proof techniques can be applied to solve mathematical problems.


  True
• Proof by induction requires both a base case and an inductive step to be valid.


  True
• What is a mathematical proof?

  A logical argument
• What are statements proven true using axioms and other theorems called?

  Theorems
• A direct proof demonstrates the truth of a statement by logically deducing the conclusion from the given premises
• In proof by contradiction, the assumption is that the original statement is false.


  True
• Indirect proof assumes the statement is false to derive a logical contradiction.

  True
• What is the primary goal of a direct proof?

  Deduce conclusion from premises
• What is the starting assumption in a proof by contradiction?

  Negation of the statement
• Proof by contradiction is useful when a direct proof is straightforward.

  False
• In the inductive hypothesis, we assume the statement is true for an arbitrary natural number $k$.


  True
• Theorems are statements proven true using axioms and other theorems.
• Axioms are fundamental statements accepted as true without proof.


  True
• The sum of two even numbers is always even.

  True
• Assume that $\sqrt{2}$ is rational, i.e., \sqrt{2} = \frac{p}{q}</latex> for integers $p$ and $q$ with no common factors
• In a proof by contradiction, the negation of the original statement must lead to a logical contradiction.


  True
• If the base case and inductive step hold in a proof by induction, the statement is true for all natural numbers.


  True
• The inductive step in a proof by induction requires proving the statement is true for $k +$$1$ based on the assumption $P(k)$.


  True
• Proof by Induction is used to prove statements for all natural numbers.
• Match the proof type with its key characteristic:

  Direct Proof ↔️ Starts with given premises.

  Indirect Proof ↔️ Starts by assuming the negation of the statement.