Core proof track
Counterexamples
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Subsections
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What Is a Counterexample?
A counterexample is a single specific instance that makes a universal statement false.
How to Spot a False Conjecture
False conjectures often fail at boundary or extreme values—testing these systematically reveals counterexamples quickly.
Building “Minimal” Counterexamples
Minimal counterexamples isolate exactly where a statement breaks, revealing structural flaws in the conjecture.
Counterexamples in Algebra & Number Theory
Algebra and number theory contain many tempting but false generalizations—counterexamples show where intuition must be corrected.
Common Patterns That Often Fail
Many common mathematical “rules” only work in certain structures—counterexamples identify exactly where patterns break.
How Counterexamples Strengthen Understanding
Counterexamples sharpen mathematical intuition by revealing exactly which assumptions matter and why.