Prove statements by showing their negation leads to impossibility
Proof by contradiction (also called reductio ad absurdum) is a powerful indirect proof technique. Instead of proving a statement P directly, we assume its negation ¬P is true and show this leads to a logical impossibility. Since mathematics cannot tolerate contradictions, our assumption must be false, meaning P must be true!
Understand the flow: assume ¬P, derive a contradiction, conclude P
Explore famous proofs like √2 irrationality and infinitude of primes
Recognize problems where contradiction is the natural approach
Avoid mistakes like circular reasoning and false contradictions
Watch how proof by contradiction flows: we assume the opposite, derive consequences, and reach an impossibility.
The contradiction proves our assumption (¬P) was false. Since ¬P is false, P must be true! This indirect approach often works when direct proof is difficult.
To prove that a statement P is true using contradiction:
This works because in logic, a statement and its negation cannot both be true.
By assuming the opposite of what we want to prove, we can often access useful information that wouldn't be available in a direct proof. The negation gives us something concrete to work with.
A genuine contradiction is something impossible: 0 = 1, a number being both even and odd, or violating a known theorem. Simply getting an "unexpected" result is not enough!
This classic proof shows that √2 cannot be written as a fraction. Watch the "lowest terms" badge as we progress!
We want to prove that √2 is irrational (cannot be written as a fraction).
Suppose we had a "complete list" of all primes. We'll construct a number N that contradicts this assumption!
Since N = (p₁ × p₂ × ... × pₖ) + 1, dividing N by any pᵢ gives remainder 1. The product part is divisible by pᵢ, but then we added 1, leaving remainder 1. This guarantees N cannot be divisible by any prime in our list!
Getting an "unexpected" or "strange" result is not a contradiction. You need something logically impossible, like proving both X and ¬X, or showing 1 = 0.
Don't assume what you're trying to prove! You can only use the assumption ¬P and previously established facts. Accidentally using P in your derivation invalidates the proof.
Keep track of your assumption ¬P throughout the proof. The contradiction should involve this assumption, either directly or through consequences derived from it.