Examples of Proof by Contradiction

1. Formula
   Observation

   The handwritten symbol $\mathbb{N}$ is used in the assumption line and later membership statements.

2. Audio
   Observation

   The speaker says he will use "the symbol N" for natural numbers.

$\mathbb{N}$

The set of natural numbers.

Natural numbers.

1. Formula
   Observation

   The handwritten line reads "let x be the last $\mathbb{N}$" followed by "$x \in \mathbb{N}$".

2. Audio
   Observation

   The speaker says, "we let x be the last natural number."

x

An arbitrary element chosen to represent the supposed largest or last natural number under the contradiction assumption.

$x \in \mathbb{N}$ under the assumption stated in the video.

1. Formula
   Observation

   The handwritten lines show "$x+1 \in \mathbb{N}$" and "$x+1 > x$".

2. Audio
   Observation

   The speaker explains that if you take a natural number and add one, you get a bigger natural number.

$x+1$

The successor of $x$, constructed to produce a larger natural number than the assumed last one.

Under the video's assumption, $x+1 \in \mathbb{N}$ whenever $x \in \mathbb{N}$.

1. Formula
   Observation

   The presenter writes `let x, y be even numbers.` and later uses `x` in the product line.

2. Audio
   Observation

   The speaker says, "So we're going to let x and y, these are our two even numbers that represent every even number."

x

An arbitrary even number used as one factor in the product.

Even number; represented later as $x=2c$ with $c\in\mathbb{N}$.

1. Formula
   Observation

   The presenter writes `let x, y be even numbers.` and later uses `y` in the product line.

2. Audio
   Observation

   The speaker says, "If y is an even number then y must also be twice some other number."

y

An arbitrary even number used as the second factor in the product.

Even number; represented later as $y=2p$ with $p\in\mathbb{N}$.

1. Formula
   Observation

   The presenter writes `$x = 2c c \in N$.`

2. Audio
   Observation

   The speaker says, "So let's say x is twice c where c is, let's say it's a natural number."

1. The video does not explicitly state whether $\mathbb{N}$ includes $0$ or starts at $1$.

c

A natural-number multiplier showing that $x$ is even.

$c\in\mathbb{N}$ as written on screen.

1. Formula
   Observation

   The presenter writes `$y = 2p p \in N$.`

2. Audio
   Observation

   The speaker says, "So we'll say y is, let's say twice p where p is a natural number."

1. The video does not explicitly state whether $\mathbb{N}$ includes $0$ or starts at $1$.

p

A natural-number multiplier showing that $y$ is even.

$p\in\mathbb{N}$ as written on screen.

1. Formula
   Observation

   The symbol $\mathbb{N}$ appears after both $c$ and $p$ in the written definitions.

2. Audio
   Observation

   The speaker calls $c$ and $p$ natural numbers.

1. The exact convention for $\mathbb{N}$ is not stated in the clip.

$\mathbb{N}$

The set of natural numbers used for the multipliers $c$ and $p$.

Number set.

1. Formula
   Observation

   $x = 2c$

x

An arbitrary even number

$x \in \mathbb{N}$

1. Formula
   Observation

   $y = 2p$

y

Another arbitrary even number

$y \in \mathbb{N}$

1. Formula
   Observation

   $c \in \mathbb{N}$

c

A natural number such that $x = 2c$

$c \in \mathbb{N}$

1. Formula
   Observation

   $p \in \mathbb{N}$

p

A natural number such that $y = 2p$

$p \in \mathbb{N}$

1. Audio
   Observation

   The speaker says, "So we're going to use a proof by contradiction, PBC."

2. Formula
   Observation

   The abbreviation "Pbc." is written on the page before the assumption line.

The video introduces the first example as a proof by contradiction, abbreviated on the page as "Pbc." The method consists of assuming the opposite of the desired statement and deriving an inconsistency.

1. The target statement is the proposition written at the top of the page: there are infinitely many natural numbers.

1. Formula
   Observation

   The symbol $\mathbb{N}$ appears in the assumption line and in "$x \in \mathbb{N}$".

2. Audio
   Observation

   The speaker says he will use "the symbol N" for natural numbers.

The video uses $\mathbb{N}$ to denote the natural numbers. In the first proof, this notation is used both in the assumption about finiteness and in membership statements such as $x \in \mathbb{N}$.

1. Used as the ambient set for the objects discussed in the first example.

1. Formula
   Observation

   The handwritten lines include "$x+1 \in \mathbb{N}$" and "$x+1 > x$".

2. Audio
   Observation

   The speaker says that if you take a natural number and add one, you get another natural number and it is bigger.

Within the first proof, the video uses the fact that from $x \in \mathbb{N}$ one may form $x+1$, which is again a natural number and satisfies $x+1>x$. This is the key construction that creates a larger natural number than the supposed last one.

1. Applied after assuming $x$ is the last natural number.

2. The video states the property informally rather than citing Peano axioms by name.

1. Notation for the natural numbers

1. Audio
   Observation

   The speaker says, "any even number can be written as twice some other number" and gives the example $10=2(5)$.

2. Formula
   Observation

   The presenter writes `$x = 2c c \in N$.` and `$y = 2p p \in N$.`

1. The clip does not discuss negative even numbers or zero explicitly.

In this clip, an even number is characterized algebraically as twice another number. The presenter applies this to the arbitrary factors $x$ and $y$, writing them as $x=2c$ and $y=2p$ with $c,p\in\mathbb{N}$.

1. The representation is applied to the arbitrary even numbers $x$ and $y$ in the proof.

2. The clip uses $\mathbb{N}$ for the multipliers without specifying whether $0\in\mathbb{N}$.

1. Audio
   Observation

   The speaker says, "we'll again we'll use a proof by contradiction. So PBC. So we're going to assume the opposite."

2. Formula
   Observation

   The presenter writes `Assume that the product of two even numbers is odd.`

The method introduced here is to prove a universal statement by assuming its opposite and seeking a contradiction. For the present proposition, the opposite of "the product of two even numbers is even" is written as "the product of two even numbers is odd."

1. Used when the goal is to prove a statement for all relevant objects rather than only checking examples.

2. The contradiction has not yet been reached within this clip.

1. Formula
   Observation

   The presenter writes `Product = xy`, then `= (2c)(2p)`, then `= 4cp`, and finally `= 2(2cp)`.

2. Audio
   Observation

   The speaker says, "Now let's get the product. So the product of our two even numbers is xy, which is the same as 2c multiplied by 2p ... And 2c by 2p is 4cp ... another way to write 4 times cp would be twice 2cp."

1. The final algebraic line is visible but the clip ends before any contradiction is derived from it.

Once $x$ and $y$ are written as even numbers, their product is transformed into a multiple of $2$: $xy=(2c)(2p)=4cp=2(2cp)$. This shows the product has the algebraic form of an even number.

1. $x=2c$ and $y=2p$ with $c,p\in\mathbb{N}$ as established earlier in the clip.

1. Definition of an even number

1. Formula
   Observation

   $x = 2c$, $c \in \mathbb{N}$

An even number is defined as any integer that can be expressed as twice another integer. In this video, it is specifically shown as $x = 2c$ where c is a natural number.

1. c must be an integer or natural number

1. Audio
   Observation

   Assume that the product of two even numbers is odd... Here's our contradiction... Therefore, the opposite of our assumption is true.

To prove a statement by contradiction, one assumes the negation of the statement is true. Then, logical deductions are made from this assumption until a mathematical contradiction is reached. The contradiction implies the initial assumption was false, thereby proving the original statement true.

1. Requires classical logic where a statement cannot be both true and false

1. Formula
   Observation

   The page heading reads "Prove that there are an infinite amount of natural numbers."

2. Audio
   Observation

   The speaker says, "we're going to prove that there's an infinite amount of natural numbers."

3. Formula
   Observation

   The conclusion line reads "=> There is an infinite amount of $\mathbb{N}$."

There are an infinite amount of natural numbers.

1. No external hypotheses are stated beyond ordinary arithmetic facts used in the proof.

2. The proof proceeds by assuming the negation: there is only a finite amount of $\mathbb{N}$.

Existential infinitude claim about the set $\mathbb{N}$; the proof assumes the opposite finite case and derives a contradiction.

1. Formula
   Observation

   The second page heading reads "Prove that the product of two even numbers is also even."

2. Audio
   Observation

   The speaker says, "We're going to prove that the product of two even numbers is also even."

1. The clip ends before any proof steps for this second claim are shown.

The product of two even numbers is also even.

1. Two numbers are even.

2. Their product is taken in the usual sense of multiplication.

Universal statement over pairs of even numbers.

1. Formula
   Observation

   The title written on the page is `Prove that the product of two even numbers is also even.`

2. Audio
   Observation

   The speaker says, "we want to prove for every even number, when you multiply any two even numbers, you get another even number."

For any two even numbers, their product is also even.

1. Take two arbitrary even numbers.

2. Their product is formed by ordinary multiplication.

Universal over pairs of even numbers.

1. Audio
   Observation

   The speaker says, "So if it's not even, it must be odd."

1. The clip does not formally define oddness or prove the parity dichotomy; it is used as an assumed fact.

If a number is not even, then it is odd.

1. The object under discussion is a number whose parity is being considered.

Implicitly universal over the numbers considered in the proof.

1. Formula
   Observation

   Prove that the product of two even numbers is also even.

For all integers x and y, if x is even and y is even, then their product xy is even.

1. x is an even integer

2. y is an even integer

Universal quantification over integers x and y

1. Formula
   Observation

   The handwritten proof shows "Pbc.", then "Assume that there is a finite amount of $\mathbb{N}$." then "let x be the last $\mathbb{N}$." then "$x \in \mathbb{N}$", "$x+1 \in \mathbb{N}$", "$x+1 > x$", and finally "=> There is an infinite amount of $\mathbb{N}$."

2. Audio
   Observation

   The speaker narrates each step: assume the opposite, choose the last natural number, form $x+1$, note it is still natural and larger, and conclude a contradiction.

1. The video does not explicitly write the logical rule of negation introduction; it states the contradiction verbally.

1. Expression

   $$\text{Assume that there is a finite amount of } \mathbb{N}.$$
   Explanation

   Begin the proof by contradiction with the negation of the desired statement.

   Justification

   Stated directly in the video as the opening move of the PBC argument.

   Shown in the video
2. Expression

   $$\text{let } x \text{ be the last } \mathbb{N}.$$
   Explanation

   If the natural numbers were finite, there would be a largest or final one; call it $x$.

   Justification

   The speaker says that a finite amount means there must be a last one.

   Shown in the video
3. Expression

   $$x \in \mathbb{N}$$
   Explanation

   Record that the chosen last element belongs to the natural numbers.

   Justification

   Written explicitly on the page and spoken aloud.

   Shown in the video
4. Expression

   $$x+1 \in \mathbb{N}$$
   Explanation

   Form the successor $x+1$ and assert that it is still a natural number.

   Justification

   The speaker says that adding one to a natural number gives another natural number.

   Shown in the video
5. Expression

   $$x+1 > x$$
   Explanation

   Compare sizes and note that $x+1$ is strictly larger than $x$.

   Justification

   The speaker says that if you add one, you get a bigger natural number.

   Shown in the video
6. Expression

   $$\text{Contradiction: } x \text{ was assumed to be the last natural number, but } x+1 \text{ is a larger natural number.}$$
   Explanation

   The existence of $x+1$ contradicts the assumption that $x$ was the final natural number.

   Justification

   Stated verbally in the video as the source of the contradiction.

   Shown in the video
7. Expression

   $$\Rightarrow \text{ There is an infinite amount of } \mathbb{N}.$$
   Explanation

   Since the assumption leads to contradiction, the original statement is concluded true.

   Justification

   The final handwritten conclusion line and the speaker's closing sentence.

   Shown in the video

The assumption that $\mathbb{N}$ is finite is impossible, so there are infinitely many natural numbers.

1. Formula
   Observation

   A new page appears with the heading "Prove that the product of two even numbers is also even."

2. Audio
   Observation

   The speaker introduces the next example and explains that product means multiply.

1. No algebraic setup, variable definitions, or proof steps for the even-product claim appear before the clip ends.

1. Expression

   $$\text{Prove that the product of two even numbers is also even.}$$
   Explanation

   The second example is announced as a new proposition to prove.

   Justification

   Read from the new page heading and spoken introduction.

   Shown in the video
2. Expression

   $$\text{product means multiply}$$
   Explanation

   The speaker clarifies the operation involved in the statement.

   Justification

   Spoken explanation near the end of the clip.

   Shown in the video

The clip sets up the second theorem but does not yet provide its proof.

1. Formula
   Observation

   The board shows the sequence `$x = 2c c \in N$.`, `$y = 2p p \in N$.`, `Product = xy`, `= (2c)(2p)`, `= 4cp`, `= 2(2cp)`. The clip ends before a contradiction is completed.

2. Audio
   Observation

   The speaker explains each substitution and simplification aloud, ending with "another way to write 4 times cp would be twice 2cp."

1. The derivation is incomplete within the provided 120-second clip because no contradiction is reached before the end.

1. Expression

   $$x=2c,\ c\in\mathbb{N}$$
   Explanation

   Represent the first arbitrary even number using the definition of evenness.

   Justification

   Definition of an even number as twice another number.

   Shown in the video
2. Expression

   $$y=2p,\ p\in\mathbb{N}$$
   Explanation

   Represent the second arbitrary even number in the same way.

   Justification

   Definition of an even number as twice another number.

   Shown in the video
3. Expression

   $$xy=(2c)(2p)$$
   Explanation

   Substitute the representations of $x$ and $y$ into the product.

   Justification

   Substitution using the previous two lines.

   Shown in the video
4. Expression

   $$(2c)(2p)=4cp$$
   Explanation

   Multiply the constants together and collect the variables.

   Justification

   Basic arithmetic and associativity/commutativity of multiplication.

   Shown in the video
5. Expression

   $$4cp=2(2cp)$$
   Explanation

   Rewrite the product as twice another quantity.

   Justification

   Factor out a 2 from $4cp$.

   Shown in the video

Within this clip, the product of the two assumed even numbers has been rewritten as $2(2cp)$, i.e. in the algebraic form of an even number; the contradiction step is not shown before the clip ends.

1. Formula
   Observation

   Assume that the product of two even numbers is odd. Let x, y be even numbers. $x = 2c$, $c \in \mathbb{N}$. $y = 2p$, $p \in \mathbb{N}$. Product = xy = (2c)(2p) = 4cp = 2(2cp). => Product of two even numbers is even.

1. Expression

   $$\text{Assume the product of two even numbers is odd.}$$
   Explanation

   State the negation of the theorem to be proved.

   Justification

   Setup for proof by contradiction.

   Shown in the video
2. Expression

   $$\text{Let } x, y \text{ be even numbers.}$$
   Explanation

   Introduce arbitrary even numbers based on the problem statement.

   Justification

   Given condition of the theorem.

   Shown in the video
3. Expression

   $$x = 2c, c \in \mathbb{N}; \quad y = 2p, p \in \mathbb{N}$$
   Explanation

   Express the even numbers using the definition of evenness.

   Justification

   Definition of an even number.

   Shown in the video
4. Expression

   $$\text{Product} = xy = (2c)(2p)$$
   Explanation

   Substitute the expressions for x and y into the product.

   Justification

   Algebraic substitution.

   Shown in the video
5. Expression

   $$= 4cp = 2(2cp)$$
   Explanation

   Simplify the algebraic expression and factor out a 2.

   Justification

   Arithmetic properties of multiplication.

   Shown in the video
6. Expression

   $$\text{Since } 2cp \text{ is an integer, } 2(2cp) \text{ is an even number.}$$
   Explanation

   Conclude that the resulting product is even because it is 2 times an integer.

   Justification

   Definition of an even number.

   Derived from the video
7. Expression

   $$\text{This contradicts the assumption that the product is odd.}$$
   Explanation

   Point out that the derived result (even) contradicts the initial assumption (odd).

   Justification

   Logical contradiction.

   Shown in the video
8. Expression

   $$\therefore \text{ the product of two even numbers is even.}$$
   Explanation

   Conclude the proof; since the assumption led to a contradiction, the original statement must be true.

   Justification

   Principle of proof by contradiction.

   Shown in the video

The product of two even numbers is always even.

1. Formula
   Observation

   The first page states the problem and then carries out the full handwritten proof.

2. Audio
   Observation

   The speaker describes this as an example of proof by contradiction and walks through the argument line by line.

Prove that there are an infinite amount of natural numbers.

1. The set under discussion is $\mathbb{N}$, the natural numbers.

2. The proof method is proof by contradiction.

3. The arithmetic fact used is that adding 1 to a natural number yields a larger natural number.

Derive a contradiction from the assumption that $\mathbb{N}$ is finite and conclude that $\mathbb{N}$ is infinite.

1. Expression

   $$\text{Assume that there is a finite amount of } \mathbb{N}.$$
   Explanation

   Start by assuming the opposite of what is to be proved.

   Justification

   Explicitly written and spoken in the video.

   Shown in the video
2. Expression

   $$\text{let } x \text{ be the last } \mathbb{N}.$$
   Explanation

   Choose a supposed largest natural number under the finite assumption.

   Justification

   The speaker says finiteness implies there must be a last one.

   Shown in the video
3. Expression

   $$x \in \mathbb{N}$$
   Explanation

   State membership of the chosen element.

   Justification

   Written on the page.

   Shown in the video
4. Expression

   $$x+1 \in \mathbb{N}$$
   Explanation

   Construct the successor and assert it remains natural.

   Justification

   Spoken and written in the proof.

   Shown in the video
5. Expression

   $$x+1 > x$$
   Explanation

   Show the successor is strictly larger.

   Justification

   Spoken and written in the proof.

   Shown in the video
6. Expression

   $$\Rightarrow \text{ There is an infinite amount of } \mathbb{N}.$$
   Explanation

   Conclude the original statement after reaching contradiction.

   Justification

   Final handwritten conclusion and spoken wrap-up.

   Shown in the video

There is an infinite amount of $\mathbb{N}$.

The contradiction is that $x$ was assumed to be the last natural number, yet $x+1$ is also a natural number and satisfies $x+1>x$.

1. Formula
   Observation

   The second page heading reads "Prove that the product of two even numbers is also even."

2. Audio
   Observation

   The speaker introduces this as another example and explains that product means multiply.

1. The clip ends before the proof begins, so no worked solution is available in this segment.

Prove that the product of two even numbers is also even.

1. Two numbers are even.

2. The operation in question is multiplication.

Show that multiplying two even numbers produces an even number.

1. Expression

   $$\text{Introduce the statement to be proved.}$$
   Explanation

   The video presents the theorem on a fresh page.

   Justification

   Visible heading and spoken introduction.

   Shown in the video
2. Expression

   $$\text{Clarify that product means multiply.}$$
   Explanation

   The speaker defines the operation verbally before any algebra starts.

   Justification

   Audio near the end of the clip.

   Shown in the video

No final answer is reached within this clip.

Not applicable because the proof is not started before the segment ends.

1. Formula
   Observation

   The presenter writes `$6 \times 8 = 48$ ✓` and `$2 \times 4 = 8$ ✓` on the right side of the page.

2. Audio
   Observation

   The speaker says, "say if I multiply six by eight, I get forty-eight ... two by four is eight ... and eight is also even so it works for that one."

Check specific products of even numbers to see whether the result is even.

1. $6$ and $8$ are even numbers.

2. $2$ and $4$ are even numbers.

Provide concrete examples supporting the general statement before proving it algebraically.

1. Expression

   $$6\times 8=48$$
   Explanation

   Compute the first example product.

   Justification

   Direct arithmetic calculation shown on screen.

   Shown in the video
2. Expression

   $$2\times 4=8$$
   Explanation

   Compute the second example product.

   Justification

   Direct arithmetic calculation shown on screen.

   Shown in the video

Both example products, $48$ and $8$, are even.

The presenter marks each example with a check mark and verbally notes that the results are even.

1. Formula
   Observation

   $6 \times 8 = 48 \checkmark$; $2 \times 4 = 8 \checkmark$; $10 = 2(5)$

Verify with specific numbers that the product of two even numbers is even.

1. Even numbers 6, 8, 2, 4, 10

Show that multiplying these even numbers results in an even number.

1. Expression

   $$6 \times 8 = 48$$
   Explanation

   Multiply 6 and 8.

   Justification

   Arithmetic calculation.

   Shown in the video
2. Expression

   $$2 \times 4 = 8$$
   Explanation

   Multiply 2 and 4.

   Justification

   Arithmetic calculation.

   Shown in the video
3. Expression

   $$10 = 2(5)$$
   Explanation

   Show that 10 is an even number by expressing it as 2 times an integer.

   Justification

   Definition of an even number.

   Shown in the video

48 and 8 are even numbers, confirming the property for these examples.

The results 48 and 8 are divisible by 2, which matches the definition of even numbers.

1. Diagram
   Observation

   A white sheet on a wooden desk shows the heading "Prove that there are an infinite amount of natural numbers." and then a handwritten proof added line by line.

2. Animation
   Observation

   A hand holding a marker writes each proof step sequentially beneath the heading.

1. White sheet of paper

2. Wooden desk background

3. Marker pen

4. Handwriting

1. The page begins with only the problem statement.

2. The writer adds "Pbc.".

3. The writer adds the assumption that there is a finite amount of $\mathbb{N}$.

4. The writer adds "let x be the last $\mathbb{N}$." and "$x \in \mathbb{N}$."

5. The writer adds "$x+1 \in \mathbb{N}$" and "$x+1 > x$."

6. The writer adds the concluding line "=> There is an infinite amount of $\mathbb{N}$."

1. The camera remains fixed overhead on the page.

2. The first problem statement stays visible at the top throughout the writing process.

The visual progression mirrors the logical structure of a proof by contradiction: assumption, construction of a counterexample within the assumption, contradiction, conclusion.

1. Animation
   Observation

   The first sheet is removed and replaced by a new sheet.

2. Diagram
   Observation

   The new page heading reads "Prove that the product of two even numbers is also even."

1. First completed proof sheet

2. Second blank-headed sheet

3. Marker pen

1. The completed first proof leaves the frame.

2. A new page with a different theorem heading appears.

3. The speaker introduces the next example but no proof lines are written before the clip ends.

1. The same overhead filming style continues.

2. The setting remains a paper-on-desk demonstration.

This visual transition marks a shift from a contradiction proof about infinitude to a new parity statement whose proof is not yet developed in the clip.

1. Diagram
   Observation

   A white sheet of paper fills the frame. The theorem statement is at the top left, numeric examples are added on the right, and the proof text is written down the left side as the clip progresses.

2. Animation
   Observation

   A hand holding a marker writes each line in sequence while the camera remains fixed overhead.

1. White paper

2. Handwritten theorem statement

3. Numeric example column on the right

4. Proof lines on the left

5. Marker pen

1. First the examples $6\times8=48$ and $2\times4=8$ are added on the right.

2. Then the contradiction assumption is written on the left.

3. Then the variable definitions $x=2c$ and $y=2p$ are added.

4. Finally the product computation is written line by line down to $=2(2cp)$.

1. The camera angle stays overhead and static.

2. The original theorem statement remains visible at the top throughout.

The visual organization separates concrete numerical checks from the general algebraic proof, making the move from examples to universal reasoning explicit.

1. Animation
   Observation

   A hand writes the steps of the proof by contradiction on a piece of paper, starting from the assumption, moving through the algebraic manipulation, and concluding with the final statement and Q.E.D.

1. Hand

2. Pen

3. Paper

4. Mathematical text

1. Text appears sequentially on the paper as the proof is constructed.

1. The camera angle remains fixed on the paper.

Visualizes the step-by-step logical construction of a mathematical proof.

1. Audio
   Observation

   The speaker says it is obvious that there are infinitely many natural numbers, "but that's not good enough for maths. We have to be able to prove that there is an infinite amount."

One might think that because a statement feels obvious, it needs no formal justification.

The video explicitly rejects that idea and insists on proving the infinitude of $\mathbb{N}$ rather than relying on intuition alone.

1. Audio
   Observation

   The speaker says, "product means multiply."

A learner may confuse the word "product" with something other than multiplication in this context.

The video directly clarifies that in the second example, "product" means the result of multiplying two numbers.

1. Audio
   Observation

   After checking examples, the speaker says, "But we want to prove for every even number, when you multiply any two even numbers, you get another even number. So we're going to have to use algebra."

One might think that verifying a few cases such as $6\times8=48$ and $2\times4=8$ proves the statement for all even numbers.

The video explicitly contrasts example checking with the need for a general algebraic proof covering every pair of even numbers.

1. Audio
   Observation

   The speaker says, "So we still haven't found any contradiction" just before rewriting $4cp$ as $2(2cp)$. The clip ends immediately afterward.

One might infer from the setup alone that the proof by contradiction is already complete.

Within the provided 120 seconds, the presenter has only rewritten the product as $2(2cp)$ and has not yet derived the contradiction with the assumption that the product is odd.

1. Audio
   Observation

   The speaker announces proof by contradiction and then immediately applies it to the natural-numbers statement.

2. Formula
   Observation

   The page shows "Pbc." followed by the assumption and contradiction steps.

The method item "proof by contradiction" is applied to establish the theorem that there are infinitely many natural numbers.

1. Formula
   Observation

   The symbol $\mathbb{N}$ appears in the assumption, membership, and conclusion lines of the first proof.

Understanding $\mathbb{N}$ as the set of natural numbers is required to read the theorem and its contradiction proof.

1. Formula
   Observation

   The lines "$x+1 \in \mathbb{N}$" and "$x+1 > x$" are used to contradict the assumption that $x$ is the last natural number.

The contradiction step depends on the successor property that $x+1$ remains in $\mathbb{N}$ and is larger than $x$.

1. Audio
   Observation

   After finishing the first proof by contradiction, the speaker says, "Let's have a look at another one," and introduces the even-product statement.

2. Diagram
   Observation

   A new page appears with a different theorem heading.

1. The clip does not show which proof method will be used for the second example.

The video contrasts two example propositions: one about infinitude of $\mathbb{N}$ proved by contradiction, and another about parity of products introduced afterward.

1. Audio
   Observation

   The speaker introduces proof by contradiction only after stating the goal of proving the proposition for every even number.

2. Formula
   Observation

   The written assumption directly negates the theorem statement.

The proof-by-contradiction method is applied to the proposition that the product of two even numbers is even.

1. Formula
   Observation

   The definitions $x=2c$ and $y=2p$ are used immediately in the product computation.

The algebraic expression for the product depends on first rewriting each factor using the definition of an even number.

1. Audio
   Observation

   The speaker moves from numeric examples to the need for algebra to prove the statement for every even number.

The numerical examples illustrate the proposition but are contrasted with the general proof required for all even numbers.

1. Formula
   Observation

   $x = 2c$ is used in the derivation Product = xy = (2c)(2p)

The definition of an even number is directly applied in the algebraic steps of the proof derivation.

1. Audio
   Observation

   The narrator explicitly states 'Here's our contradiction' and 'the opposite of our assumption is true' during the derivation.

The general method of proof by contradiction is instantiated in the specific derivation shown.

1. Audio
   Observation

   The speaker names the method as proof by contradiction and abbreviates it as PBC.

1. Proof by contradiction as the method for the first example

1. Formula
   Observation

   The full handwritten proof is visible on the first page.

1. Proof by contradiction as the method for the first example
2. Notation for the natural numbers
3. Adding one to a natural number stays in $\mathbb{N}$ and increases size
4. There are infinitely many natural numbers
5. Proof that there are infinitely many natural numbers by contradiction
6. Example 1: Prove that there are infinitely many natural numbers

1. Audio
   Observation

   The speaker explains that $x+1$ is a bigger natural number, contradicting the assumption that $x$ was last.

1. Adding one to a natural number stays in $\mathbb{N}$ and increases size
2. Proof that there are infinitely many natural numbers by contradiction

1. Formula
   Observation

   The second page heading introduces the even-product theorem.

2. Audio
   Observation

   The speaker says they are going to prove it but the clip ends before the proof starts.

1. The product of two even numbers is even
2. Example 2: Prove that the product of two even numbers is even
3. Introduction to the second example without completed derivation

1. Audio
   Observation

   The speaker explicitly says examples are not enough and algebra is needed to prove the statement for every even number.

1. Numerical examples illustrating the claim
2. Proposition to be proved
3. Proof by contradiction setup

1. Formula
   Observation

   The written line `Assume that the product of two even numbers is odd.`

2. Audio
   Observation

   The speaker says they will assume the opposite and notes that if it is not even it must be odd.

1. Proof by contradiction setup
2. Proposition to be proved
3. Parity dichotomy used in the contradiction assumption

1. Formula
   Observation

   The board shows `$x = 2c c \in N$.` and `$y = 2p p \in N$.`

2. Audio
   Observation

   The speaker says any even number can be written as twice some other number.

1. Definition of an even number
2. x
3. y
4. c
5. p

1. Formula
   Observation

   The product is written as `xy = (2c)(2p) = 4cp = 2(2cp)`.

1. Algebraic form of the product of two even numbers
2. Derivation of the product form under the contradiction assumption

1. Formula
   Observation

   Prove that the product of two even numbers is also even.

1. Derivation of the Proof by Contradiction
2. Theorem: Product of Two Even Numbers is Even

1. Audio
   Observation

   Assume that the product of two even numbers is odd... Here's our contradiction...

1. Method of Proof by Contradiction

1. Formula
   Observation

   $x = 2c$, $c \in \mathbb{N}$

1. Definition of an Even Number