Mathematical Proofs

Why This Chapter Matters

Mathematics is not only about getting correct answers; it is about knowing why those answers are correct. A mathematical proof is a precise, logical explanation that shows a statement must be true, starting from agreed-upon assumptions (axioms, definitions, or previously proved results).

This chapter sets the stage for the more detailed chapters on specific proof techniques and on writing proofs well. Here, the focus is on:

• What a mathematical proof is (and is not),
• The basic ingredients of a proof,
• How proofs fit into the structure of mathematics,
• Common beginner misunderstandings about proofs,
• How to start thinking in a “proof-oriented” way.

Later chapters on “Proof Techniques” and “Writing Proofs” will go into specific methods and style.

What Is a Mathematical Proof?

A mathematical proof is a logical argument that shows a statement is necessarily true, assuming:

• Certain starting points (axioms, definitions, previously established theorems),
• Rules of logical reasoning.

A proof does not rely on:

• Repeated experiments or measurements,
• “It works for many examples”,
• Authority (“a famous mathematician said so”).

Instead, it shows that if the assumptions are true, the conclusion must be true, with no exceptions.

You can think of a proof as a chain:

$$
\text{Assumptions} \;\Rightarrow\; \text{Step 1} \;\Rightarrow\; \text{Step 2} \;\Rightarrow\; \dots \;\Rightarrow\; \text{Conclusion}
$$

Each arrow $\Rightarrow$ must follow valid logical rules.

The Building Blocks of Proofs

Although the details of logic appear in other chapters, it is helpful to recognize the main components you will see inside proofs:

• Definitions: These tell you exactly what a word means in the mathematical context. For example, “even integer” or “prime number” has a precise definition.
• Axioms (or postulates): Basic assumptions we accept without proof inside a given mathematical system (for example, in geometry or arithmetic).
• Previously proved results (theorems, lemmas, corollaries):
• Theorem: A major result.
• Lemma: A supporting result, often proved on the way to a theorem.
• Corollary: A result that follows quickly from a theorem.

A proof uses these ingredients plus rules of logical reasoning to show a new statement is true.

The Role of Proofs in Mathematics

Proofs serve several important purposes:

1. Certainty
   They provide absolute assurance (within the accepted axioms and logic) that a statement is true, not just “probably true”.
2. Understanding
   A good proof does more than verify truth; it explains why something is true, often revealing structure and connections between ideas.
3. Organization
   Proofs link results together. Mathematics builds up from basic facts to more complex theories, with proofs forming the connections.
4. Communication
   Proofs are how mathematicians convince each other that a new claim is correct. They form a common language of justification.

Types of Statements You May Prove

In this course, you will encounter several kinds of statements that can be proved. The detailed logical forms are handled elsewhere; here is a simple overview.

1. Universal statements
   These say something is true for all objects of a certain kind.
• General form: “For all $x$ with property P, $Q(x)$ is true.”
• Example idea: “For all even integers $n$, $n^2$ is even.”
2. Existential statements
   These assert that there exists at least one object with a certain property.
• General form: “There exists $x$ such that $P(x)$ is true.”
• Example idea: “There exists an irrational number whose square is rational.”
3. Conditional statements (implications)
   These have the form “If A, then B.”
• Example idea: “If a number is divisible by 4, then it is even.”
• Often written as $A \Rightarrow B$.
4. Equivalences (“if and only if”)
   These say two conditions always occur together:
• General form: “A if and only if B.”
• Written as $A \Leftrightarrow B$.
• This actually combines two implications: $A \Rightarrow B$ and $B \Rightarrow A$.
5. Negations and contradictions
   Sometimes you will prove that a certain statement cannot be true, or that assuming it leads to a contradiction. These are central to some proof techniques and are treated in more detail later.

What Proofs Are Not

Especially for beginners, it is important to distinguish proofs from other kinds of reasoning.

1. Proof is not checking many examples
   Verifying that something works for many specific cases (for example, plugging in numbers) can suggest a pattern, but it does not guarantee it always holds.
• Example: Checking that $2^n - 1$ is prime for $n = 2,3,5,7$ does not prove it is prime for all $n$.
2. Proof is not a picture alone
   A diagram can suggest a relationship and help you see what is going on, but it is not a proof unless its conclusions are justified step by step.
3. Proof is not “it seems obvious”
   Many statements that seem “obvious” can turn out to be false, and many true statements are far from obvious. Mathematics requires precise justification.
4. Proof is not an appeal to authority
   Saying “my teacher/The book says so” is not a mathematical proof. The point is to show the reasoning, not just the conclusion.

The General Shape of a Proof

Although specific techniques differ, many proofs follow a similar overall structure:

1. State what you are proving
   Clearly specify the theorem or statement.
2. Recall relevant definitions or assumptions
   Identify what each term in the statement means and what is given.
3. Develop a logical argument
   Move step by step from given information and known results toward the desired conclusion.
4. Make the conclusion explicit
   End by clearly indicating you have shown exactly what was required.

In written form, proofs often begin with something like “Proof:” and end with a marker such as a small square $\square$ (sometimes called a “QED symbol”) to show that the argument is complete.

Proofs in Different Areas of Mathematics

The style of proofs often reflects the area of mathematics:

• Algebra: Proofs may manipulate equations and inequalities using algebraic rules.
• Number theory: Proofs often involve divisibility, modular arithmetic, and properties of integers.
• Geometry: Proofs may rely on diagrams for intuition but must rest on geometric definitions and postulates.
• Analysis (calculus and beyond): Proofs can involve limits, inequalities, and careful handling of “small quantities.”
• Linear algebra: Proofs may involve properties of vectors, matrices, and linear transformations.

Despite different appearances, all of these use the same underlying idea: step-by-step logical reasoning from clearly stated assumptions.

Common Beginner Difficulties

When you first learn to work with proofs, it is normal to encounter certain challenges:

1. Not using definitions precisely
   Many proofs depend on writing out and using definitions exactly. Skipping this step often causes confusion.
2. Assuming what you need to prove
   A frequent error is to start a proof by writing the conclusion and working “backwards” without justification, unintentionally using what you are trying to prove as if it were already known.
3. Leaping over steps
   Omitting intermediate reasoning can hide mistakes and make it hard for others (and yourself) to check correctness.
4. Confusing examples with proofs
   Providing one or two examples that satisfy the statement does not prove it holds always.

Recognizing these common issues makes it easier to avoid them as you practice.

Developing a Proof Mindset

To become comfortable with proofs, it helps to adjust how you think about mathematics:

• Ask “why?” not just “how?”
  Instead of only learning methods to get answers, ask why those methods work.
• Practice translating words into precise statements
  Many proofs begin with rewriting a verbal statement in a clear mathematical form.
• Expect to revise
  Proofs are often discovered through trial, error, and rethinking. The polished version you write is usually not the first attempt.
• Use small examples for insight, not as final evidence
  Trying examples can guide your thinking and suggest patterns, but then you must justify the pattern in general.

Where We Go From Here

This chapter has outlined the overall idea and role of mathematical proofs. In the chapters that follow:

• Proof Techniques will describe common methods for constructing proofs (such as direct proof, proof by contradiction, and mathematical induction).
• Writing Proofs will focus on how to present proofs clearly and effectively, including structure and style.

As you move into those chapters, keep in mind the central idea introduced here: a proof is a clear, logical explanation that shows a mathematical statement must be true, starting from agreed-upon assumptions.