Disproof of existence
The proof of Fermat’s Last Theorem is a disproof of existence; it shows that there are NO solutions to the Fermat equation.
The disproof of a statement
“There exists \(x\in S\) such that \(P(x)\)”
requires proving a universal statement:
“For all \(x\in S\), not \(P(x)\).”
Disproof of existence
Claim: There exists a pythagorean triple \((a,b,c)\) such that all of \(a\), \(b\), and \(c\) are odd.
The negation of this claim is
“For all pythagorean triples \((a,b,c)\), at least one of \(a\), \(b\), or \(c\) is even.”
Disproof of existence by contradiction
Proof by contradiction is often useful to prove “nonexistence” of something.
Claim: There is a real number \(x\) such that \(x\in(x^4, x^2)\). (See Example 9.5).