Proof and counterexample: knowing what evidence establishes

A universal statement says that every allowed object satisfies a condition. An existential statement says that at least one does. To establish existence, constructing one valid example is sufficient. To refute a universal statement, constructing one valid counterexample is sufficient. The example must satisfy the original assumptions; a negative number cannot refute a statement restricted to positive numbers, even if it breaks the proposed conclusion.

Many favorable examples do not normally prove a universal statement over an infinite domain. Testing the first thousand integers leaves all later integers untested. Finite exhaustive checking can be a proof when the domain is genuinely finite and every case is rigorously covered, but a floating-point sample or a plotted interval is rarely exhaustive in that sense. State exactly which domain your evidence covers.

To prove that the square of an odd integer is odd, begin with the definition: an odd integer can be written as $2k+1$ for an integer $k$. Expand its square and factor out a two from all but the last term. The resulting form is again twice an integer plus one. The proof covers every odd integer because $k$ was arbitrary, not because one particular square looked correct.

Notice the work done by a small assumption: the expression inside the final parentheses must be an integer. If the original input were a general real number, “odd” would not have the same definition. A good proof names its arbitrary object, uses only licensed operations, and finishes by reconnecting the derived expression to the desired definition. Clear prose matters as much as symbolic manipulation.

An implication says that if its premise holds, its conclusion must hold. It does not automatically establish the converse. For example, being divisible by four implies being even, but being even does not imply divisibility by four: six is a counterexample. Reversing an arrow requires its own argument. Likewise, a proof under idealized assumptions does not establish that a real instrument or production service satisfies those assumptions.

Use computation as a partner in reasoning. It can suggest a conjecture, reveal an overlooked edge case, or verify arithmetic inside a proof. It cannot by itself turn finitely sampled evidence into an unrestricted theorem. In advanced study you will encounter controlled approximation errors and probabilistic guarantees. Those are meaningful conclusions when their conditions are explicit; they should not be silently promoted to exact or unconditional claims.