Disproof by Counter Example

A universal statement is a mathematical claim asserted to be true for every element within a specific set or domain, often using phrases like 'for all' or 'for every'. In formal logic, this is represented by the universal quantifier $\forall$.

A counter-example is a specific case or value within the defined domain that contradicts the general claim. If a statement claims that a property $P(x)$ is true for all $x$, a counter-example is any $x$ such that $P(x)$ is false.

The power of this method lies in its simplicity: while proving a universal statement might require complex deduction or exhaustion, disproving it only requires a single, well-chosen example. This is because the logical negation of 'for all $x$, $P(x)$ is true' is 'there exists at least one $x$ such that $P(x)$ is false' ($\exists x, \neg P(x)$).

Step 1: Identify the Domain

• Carefully determine the set of numbers the statement applies to (e.g., natural numbers $\mathbb{N}$, integers $\mathbb{Z}$, or real numbers $\mathbb{R}$). A counter-example must be a member of this specific set.

Step 2: Test 'Unusual' Values

• Many false conjectures fail at boundary points or with numbers that have unique properties. Common candidates include $0$, $1$, negative numbers, or the only even prime number, $2$.

Step 3: Verify the Failure

• Substitute the chosen value into the statement and show clearly that the resulting condition is false. For example, if the statement involves an inequality, show that the left-hand side is not actually greater than the right-hand side for your chosen value.

• Check the Domain Constraints: Examiners often provide statements that are true for positive integers but false for negative integers or fractions. Always verify if your counter-example belongs to the set defined in the question.

• The 'Number 2' Trick: In questions involving prime numbers, the number $2$ is the most frequent counter-example because it is the only even prime. Most general statements about primes assume they are odd, which $2$ refutes.

• Trigonometric Boundaries: For statements involving $\sin(x)$ or $\cos(x)$, check values like $0, \frac{\pi}{2}, \pi$ or angles where the functions result in $0$ or $1$. These are common areas where general algebraic patterns in trig identities might fail.

• Clarity of Presentation: When providing a counter-example, state the value clearly, show the calculation, and conclude with a statement like 'Since this case is false, the general statement is disproven'.

• Using Values Outside the Domain: A common error is providing a counter-example that isn't allowed by the statement. For instance, using $x = -1$ to disprove a claim about 'all natural numbers' is invalid because $-1$ is not a natural number.

• Thinking One is Not Enough: Students often feel they need to provide multiple examples to 'really' disprove a point. In formal logic, one valid counter-example is just as powerful as a million.

• Confusing 'False' with 'Not Always True': In everyday language, we might say a rule is 'mostly true'. In mathematics, 'mostly true' is simply 'false'. There is no middle ground for universal statements.