1.1 Proof

• What is a proof in mathematics?

  Logical and rigorous argument
• The necessity of a proof lies in its ability to provide certainty and eliminate doubt
• Definitions in a proof must be precise and unambiguous.
• Basic assumptions in a proof are called axioms
• What is the sum of two even numbers, $a$ and $b$, if $a =$$2m$ and $b =$$2n$?


  $2(m + n)$
• Since m + n</latex> is an integer, $2(m + n)$ is an even number
• What three elements are used to construct a mathematical proof?

  Definitions, axioms, reasoning
• A proof in mathematics establishes truths based on deduction rather than empirical observation.
• What are precise statements of mathematical concepts called?

  Definitions
• Basic assumptions accepted as true without proof are called axioms
• The sum of two even numbers is always even.
• What is the starting point of a direct proof?

  Known assumptions
• Proof by contradiction begins by assuming the negation of the statement
• What two steps are required in proof by induction?

  Base case, inductive step
• Proof by induction proves that a statement holds for all numbers greater than or equal to the base case.
• Provide an example of a direct proof.

  Sum of two even numbers
• To prove that $\sqrt{2}$ is irrational, we use proof by contradiction
• What is the formula for the sum of the first $n$ natural numbers proven by induction?


  $\frac{n(n + 1)}{2}$
• The sum of two even numbers is always even, as proven by direct proof.
• What is the approach in a direct proof?

  Derive the conclusion directly
• In a direct proof, the conclusion is derived from known assumptions
• Proof by contradiction assumes the negation of the statement.
• What is the conclusion in a proof by contradiction?

  A contradiction validating the statement
• In proof by induction, the statement holds for all numbers greater than or equal to the base case.
• Steps to prove "if $a$ and $b$ are even, then $a +$$b$ is even"


  1️⃣ Let $a =$$2m$ and $b =$$2n$ for integers $m$ and $n$

  2️⃣ $a +$$b =$$2m +$$2n =$$2(m + n)$

  3️⃣ $a +$$b$ is even
• The sum of two even numbers is always even.
• What are the two main types of indirect proof?

  Proof by contradiction and contraposition
• In indirect proof, the approach starts with the negation of the conclusion
• Steps in a proof by contradiction

  1️⃣ Assume the statement is false

  2️⃣ Use logical steps to derive a contradiction

  3️⃣ Conclude that the original statement must be true
• What is the contradiction in the proof that $\sqrt{2}$ is irrational?


  Both $p$ andq</latex> are even
• $\sqrt{2}$ is an irrational number.
• What is the first step in a proof by contraposition?

  Form the contrapositive
• The key difference between direct and indirect proof lies in their starting point
• What is the contrapositive of "if $n^{2}$ is even, then $n$ is even"?


  If $n$ is odd, then $n^{2}$ is odd
• Steps to prove "if $n^{2}$ is even, then n</latex> is even" using contraposition


  1️⃣ Form the contrapositive: "if $n$ is odd, then $n^{2}$ is odd"

  2️⃣ Let $n =$$2k +$$1$ for some integer $k$

  3️⃣ $n^{2} =$$(2k + 1)^{2} =$$4k^{2} +$$4k +$$1 =$$2(2k^{2} +$$2k) +$$1$

  4️⃣ $n^{2}$ is odd

  5️⃣ Therefore, if $n^{2}$ is even, then $n$ is even
• A mathematical proof must be based on axioms, definitions, and previously established theorems.
• What are the key elements of a mathematical proof?

  Definitions, axioms, logical reasoning
• Basic assumptions accepted as true without proof are called axioms
• A structured sequence of steps to derive a conclusion is known as logical reasoning
• An even number can be written as $2k$, where $k$ is an integer.