John O'Brien · English · ExplanationConnection evidence
Reviewed English material v1: card "Proof by Contradiction Method" at 4:00 — A logical method where you assume the opposite of what you want to prove. If this assumption leads to a mathematical contradiction, the original assumption is false, proving your initial statement true.
Real Analysis Summer 2020 - Max Wimberley · English · ProofConnection evidence
Reviewed English material v1: card "Proof by Contradiction: Path-connected to Connected" at 5:25 — The proof assumes the set E is disconnected by open sets U1 and U2. It finds points s and t in different sets, uses path-connectedness to create a path γ between them, and then shows this path's image must be both connected (by a previous theorem) and disconnected (by U1 and U2), yielding a contradiction.
Math For Life · English · ProofConnection evidence
Reviewed English material v1: card "Proving Non-Uniform Convergence by Contradiction" at 4:28 — To prove that fn(x)=xn does not converge uniformly on [0,1], we assume the opposite: that it does converge uniformly. Since each fn(x) is continuous, the theorem implies the limit function f(x) must be continuous. However, we know the pointwise limit f(x) is discontinuous at x=1. This contradiction means the assumption of uniform convergence is false.
Center of Math · English · ApplicationConnection evidence
Reviewed English material v2: card "Contradiction setup for the union theorem" at 3:05 — The proof assumes X∪Y is disconnected and introduces open sets U and V that separate it. The board writes U∪V=X∪Y as the starting point for the contradiction.
Questions and answers about Proof by contradiction
Constructing x+1 creates a contradiction because the proof initially assumes that x is the last natural number. However, by adding 1 to x, we get x+1, which is still a natural number and is strictly larger than x. This directly contradicts the assumption that x was the final or largest natural number, proving that the set of natural numbers cannot be finite.
The proof assumes there is a finite amount of N.
x is defined as the last natural number under this assumption.
The arithmetic property used is that if x∈N, then x+1∈N and x+1>x.
Example: The speaker explains that if you take a natural number and add one, you get a bigger natural number. The written lines show "x+1∈N" and "x+1>x", which contradict the earlier statement "let x be the last N."
The video proves that there are infinitely many natural numbers by using a proof by contradiction. It assumes the opposite—that there is a finite amount of natural numbers—and then shows that this assumption leads to a logical contradiction. By constructing a larger natural number from the supposed last one, the proof demonstrates that the set of natural numbers cannot be finite.
The proof method is proof by contradiction (PBC).
The set under discussion is N, the natural numbers.
The arithmetic fact used is that adding 1 to a natural number yields a larger natural number.
Example: The handwritten proof shows: "Pbc.", "Assume that there is a finite amount of N.", "let x be the last N.", "x∈N", "x+1∈N", "x+1>x", and finally "=> There is an infinite amount of N."
For the theorem that the product of two even numbers is even, the proof by contradiction assumes the opposite: that the product of two even numbers is odd. The speaker justifies this by stating that if a number is not even, it must be odd. This assumption sets up the logical framework to derive a contradiction.
The theorem to be proved is: the product of two even numbers is also even.
The method is proof by contradiction.
The parity dichotomy (not even implies odd) is used as an assumed fact.
Example: The presenter writes "Assume that the product of two even numbers is odd." The speaker says, "we'll again we'll use a proof by contradiction. So PBC. So we're going to assume the opposite." and "So if it's not even, it must be odd."
In this video, PBC stands for "proof by contradiction." It is a mathematical method used to prove a statement by assuming the opposite of the desired statement and deriving an inconsistency or contradiction. The video explicitly abbreviates "proof by contradiction" as "Pbc." on the page before starting the first example.
The context is a mathematical proof.
The method involves assuming the negation of the target statement.
Example: The speaker says, "So we're going to use a proof by contradiction, PBC." The abbreviation "Pbc." is written on the page before the assumption line.