Mathematical proofs explain why a statement is always true, not just why it works in a few examples. This cheat sheet covers three major proof methods: direct proof, proof by contradiction, and mathematical induction. Students need these methods to justify algebra, number theory, geometry, sequences, and later advanced mathematics.
A clear proof structure helps turn ideas into logical arguments.
Key Facts
- A direct proof starts with the given information and uses definitions, algebra, and known facts to reach the conclusion.
- To prove a conditional statement directly, assume is true and show that must be true for the statement .
- A proof by contradiction assumes the negation of the conclusion, then shows that this assumption leads to an impossible statement such as .
- The contradiction method proves by assuming and deriving both and , which cannot both be true.
- A mathematical induction proof has a base case, usually or , and an inductive step from to .
- In induction, the induction hypothesis assumes the statement is true for , and the goal is to prove it true for .
- A common induction template is: prove , assume , prove , then conclude is true for all integers .
- To disprove a universal statement such as , it is enough to find one counterexample where is false.
Vocabulary
- Direct proof
- A proof method that begins with known facts or assumptions and logically derives the desired conclusion.
- Contradiction
- A proof method that assumes the opposite of what must be proven and shows that this assumption creates an impossibility.
- Mathematical induction
- A proof method used to prove statements about integers by proving a starting case and a repeating step.
- Base case
- The first value in an induction proof, such as , that must be verified directly.
- Induction hypothesis
- The temporary assumption in an induction proof that the statement is true for .
- Counterexample
- A single example that proves a universal statement is false.
Common Mistakes to Avoid
- Proving only examples, then claiming the statement is always true, is wrong because several true cases do not prove a universal rule.
- Forgetting the base case in induction is wrong because the inductive step only shows a chain continues after it has started.
- Using the conclusion as a reason inside a direct proof is circular reasoning because it assumes what the proof is supposed to establish.
- Assuming in an induction proof is wrong because the induction hypothesis only allows you to assume .
- Stopping a contradiction proof after making the assumption is wrong because the proof must actually derive an impossible result or a statement known to be false.
Practice Questions
- 1 Use a direct proof to show that if is an even integer, then is even.
- 2 Use proof by contradiction to show that is irrational.
- 3 Use mathematical induction to prove that for all integers .
- 4 Explain when induction is a better proof strategy than checking many numerical examples.
Understanding Mathematical Proofs (Direct, Contradiction, Induction)
A proof begins by reading the claim with care. Students should identify the objects under discussion and the conditions placed on them. A claim about integers may fail for fractions.
A claim about positive numbers may fail at zero. Definitions matter because they give the exact meaning needed for the argument. For example, an even integer is a number that can be written as two times an integer.
That definition gives a useful starting form. In a direct argument, every line should follow from an earlier line for a stated reason. A calculation alone is not enough if the reader cannot see why each substitution or simplification is allowed.
Contradiction is useful when the opposite assumption has strong consequences. A classic example concerns the square root of two. Suppose it could be written as a fraction in lowest terms.
Squaring the fraction shows that the numerator must be even. This forces the denominator to be even too. The fraction was supposed to be in lowest terms, so this is impossible.
The important point is that the impossible result comes from the temporary assumption, not from the original claim. Students often make a weak contradiction by reaching a surprising result. A real contradiction must break a definition, a known fact, or an earlier condition.
Induction depends on the order of the counting numbers. Think of a row of linked steps. The first step must be supported, and the rule must show that support moves from any step to the next one.
The inductive assumption is a tool, not a fact proved from nothing. It may be used only for the particular number named in the step. Then the work must connect that case to the next number.
Many errors come from proving a statement about the same number twice, or from using the result for the next number before it has been established. Some problems need strong induction.
In that form, the argument may use every earlier case to establish the next case. This appears in problems about factoring integers or sequences built from earlier terms.
Proof writing is a skill of communication as well as logic. A good proof names its assumptions, keeps variables consistent, and ends by clearly linking the work to the claim. Diagrams can suggest ideas in geometry, but labels and visual appearance do not count as proof by themselves.
Testing small cases is valuable because it can reveal a false formula or an indexing mistake. It cannot establish a rule for infinitely many cases. One failed case is enough to reject a statement that claims something happens for every value.
These habits matter beyond classwork. Computer programs, encryption methods, and scientific models all rely on careful reasoning about what follows from stated assumptions.