A mathematical proof is a clear argument that shows a statement must be true, not just that it seems true from examples. Proof methods are tools for organizing reasoning so each step follows from definitions, known facts, or earlier results. Learning these methods matters because proof is the language used to justify results in algebra, geometry, calculus, number theory, and computer science.
Good proofs also build habits of precise thinking and careful communication.
Different proof techniques fit different kinds of statements. A direct proof starts with the assumptions and moves step by step to the conclusion, while a proof by contrapositive proves an equivalent statement instead. A proof by contradiction assumes the statement is false and shows that this leads to an impossibility.
Counterexamples are used to disprove universal claims by finding just one case where the claim fails.
Understanding Math: Methods of Proof
A conditional claim has two parts. The hypothesis gives the starting condition. The conclusion gives the result that must follow.
Before choosing a method, translate every word into its exact definition. For example, saying an integer is even means it can be written as two times some integer. That hidden integer is important because it gives you something concrete to work with.
If a problem says two numbers are even, give each number its own whole-number multiplier. Do not silently assume the multipliers are the same. Much of proof writing is careful control of what has been given and what still needs justification.
Direct reasoning works best when the hypothesis supplies a useful form immediately. To show that the sum of two even integers is even, write the first as two times one integer and the second as two times another integer. Their sum becomes two times the sum of those integers.
The sum inside the parentheses is an integer, so the required definition has been reached. Notice that the final sentence matters. A proof is not complete merely because an expression looks even.
You must identify the integer that makes the definition work. Algebra in a proof has a purpose. Each rearrangement should move the expression closer to the exact meaning of the conclusion.
The contrapositive is especially useful when the original conclusion is difficult to handle, but its failure has a clear meaning. Suppose a claim says that if an integer has a certain property, then its square has another property. Instead of beginning with the first property, begin by assuming the square does not have the claimed property.
Then show that the original integer cannot have had the first property. This method often appears in divisibility and parity problems. It is vital to reverse both parts correctly.
The contrapositive of a statement about a starting condition leading to a result is the failure of the result leading to the failure of the starting condition. Reversing the order without changing to failures produces a different claim and may be false.
A contradiction proof needs a genuine impossibility, not merely an unexpected result. Common contradictions include an integer being both even and odd, a positive number being less than zero, or a number being forced to equal two different values. Keep track of the extra assumption introduced at the start, since the contradiction shows that this assumption cannot stand.
Counterexamples require a different habit. For a claim about every object in a set, test simple cases near boundaries, such as zero, one, negative values, or special shapes. One valid counterexample ends a universal claim.
A collection of supporting examples never proves it. When checking any proof, ask whether every claim follows from a definition, an earlier result, or a clearly stated calculation.
Key Facts
- Direct proof pattern: Assume P is true, use definitions and theorems, then prove Q is true.
- Contrapositive equivalence: P implies Q is logically equivalent to not Q implies not P.
- Contradiction pattern: To prove P, assume not P and derive a contradiction such as A and not A.
- Counterexample rule: A universal statement is false if one example satisfies the hypothesis but not the conclusion.
- Even integer definition: n is even means n = 2k for some integer k.
- Odd integer definition: n is odd means n = 2k + 1 for some integer k.
Vocabulary
- Proof
- A proof is a logical argument that establishes a mathematical statement using definitions, accepted facts, and valid reasoning.
- Direct proof
- A direct proof shows P implies Q by starting with P and deriving Q through valid steps.
- Contrapositive
- The contrapositive of P implies Q is not Q implies not P, and it always has the same truth value as the original statement.
- Contradiction
- A contradiction is a statement that cannot be true, such as saying a number is both even and not even.
- Counterexample
- A counterexample is a single example that proves a universal claim is false.
Common Mistakes to Avoid
- Testing only a few examples and calling it a proof is wrong because examples can support a pattern but cannot prove it for all cases.
- Proving the converse instead of the original statement is wrong because Q implies P is not logically equivalent to P implies Q.
- Using contradiction without clearly stating the negation is wrong because the reader must know exactly what assumption leads to the impossibility.
- Giving a counterexample that does not satisfy the hypothesis is wrong because it cannot disprove the claimed implication.
Practice Questions
- 1 Use a direct proof to show that if n is an even integer, then n^2 is even.
- 2 Use proof by contrapositive to prove that if n^2 is odd, then n is odd.
- 3 A student claims that if a number is divisible by 6, then it is divisible by 12. Explain whether this statement is true or false, and justify your answer using an appropriate proof method.