Proof — Definition, Formula & Examples
A proof is a logical argument that demonstrates why a mathematical statement must be true. It starts from known facts (axioms, definitions, or previously proven results) and uses valid reasoning to arrive at the statement being proved.
A proof is a finite sequence of deductive steps, each justified by an axiom, definition, or previously established theorem, that establishes the truth of a given proposition with certainty.
How It Works
To write a proof, you begin by clearly stating what you want to prove. Then you identify your starting assumptions — these are axioms, definitions, or theorems you already know to be true. Each subsequent step must follow logically from the previous ones, with no gaps in reasoning. Common proof techniques include direct proof (assume the hypothesis, derive the conclusion), proof by contradiction (assume the opposite of what you want to prove and reach an impossibility), and proof by contrapositive (prove the contrapositive of a conditional statement instead). The proof ends when you have logically arrived at the statement you set out to establish.
Example
Problem: Prove that the sum of two even integers is always even.
State assumptions: Let a and b be even integers. By definition of even, there exist integers m and n such that:
Add and simplify: Compute the sum and factor out 2:
Conclude: Since m + n is an integer, the sum a + b equals 2 times an integer, which is even by definition.
Answer: The sum of two even integers is even, because it can always be written as 2 times an integer.
Why It Matters
Proof is the foundation of all mathematical knowledge — it is how mathematicians verify that results are true beyond any doubt. You will encounter proofs in geometry, discrete math, linear algebra, and analysis. Fields like computer science and cryptography rely on proof-based reasoning to guarantee that algorithms and security protocols work correctly.
Common Mistakes
Mistake: Using a few numerical examples as proof (e.g., checking 2 + 4 = 6 and 6 + 8 = 14 are even, then claiming the statement is proven).
Correction: Examples can support a conjecture but cannot prove a universal statement. A proof must use general reasoning that covers every possible case, not just specific numbers.