Disproving a Conjecture by Providing a Counterexample

Understanding a Conjecture

A conjecture is a proposition made on the basis of some observed patterns or relationships, which is not proven yet.
It is a kind of hypothesis or educated prediction, often arising in the field of mathematics and sciences.
In mathematics, a conjecture is considered true only if it is proved for all possible cases that it applies to.

The Role of a Counterexample

A counterexample is a particular instance or case that proves a statement or conjecture false.
The use of a counterexample is a standard method for disproving conjectures in mathematics.
To disprove a conjecture, you only need one valid counterexample.

Identifying a Counterexample

First, understand what the conjecture proposes. Comprehend its variables, relations, and conditions.
A counterexample to a statement is a particular case where the statement is not true.
Not all cases that contradict a conjecture can be valid counterexamples – to be valid, a counterexample must meet the same conditions set by the conjecture itself.

Application in Disproving Conjecture

For a general statement, check it for different values of variables to identify any counterexample.
In geometrical conjectures, consider unusual shapes or configurations that the conjecture may not have accounted for.
A counterexample must be specific and clearly demonstrated. Generally, it should give all necessary details to allow someone else to replicate it.

Example

To illustrate, consider the conjecture that “All prime numbers are odd.” This seems true if we consider the first few prime numbers like 3, 5, 7, 11, and 13. However, the prime number 2 is even, not odd. Hence, 2 serves as a counterexample that disproves the conjecture.

Note: Mastering the method of disproving a conjecture using counterexamples is crucial because it helps in eliminating wrong ideas and hypotheses, and strengthens the foundation and integrity of mathematical statements that remain true in all tested cases.