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Direct proof is the most straightforward style of mathematical proof: you start from the assumptions and, step by step, use correct reasoning to arrive at the statement you want to prove.
In this chapter, “assumptions” refers to the hypotheses of the statement you’re proving (for example, “$n$ is an even integer”), plus any previously proved results and accepted definitions. The goal is to show that, under those assumptions, the conclusion must be true.
The basic pattern of a direct proof
Most direct proofs of statements of the form
$$\text{If } P \text{, then } Q$$
follow the same pattern:
- Assume $P$ is true.
You clearly state that you are working under the assumption that the hypothesis holds. - Use definitions and known results.
You rewrite $P$ and any new expressions using definitions (for example, “$n$ is even” means “there exists an integer $k$ such that $n = 2k$”) and use previously proved theorems, algebra, arithmetic, or logic as needed. - Reason step by step.
Each step follows logically from earlier steps. You do not jump to the conclusion; you show each small step that leads toward $Q$. - Arrive at $Q$.
At the end, you have a statement that is exactly (or obviously equivalent to) the desired conclusion $Q$. - Conclude the proof.
You note that you have shown $Q$ under the assumption that $P$ is true, so the implication “if $P$, then $Q$” holds.
The structure in words often looks like:
- “Let $n$ be an integer such that …”
- “By definition of … we have …”
- “Therefore …”
- “Hence, … as required.”
Common statement forms suited to direct proof
Direct proof is naturally used with some typical kinds of statements. Here are three very common forms and how they are set up.
1. Universal implication: “For all … if … then …”
Example form:
$$\forall x \in S,\; \bigl(P(x) \Rightarrow Q(x)\bigr).$$
To prove this directly:
- Start with: “Let $x$ be an arbitrary element of $S$ such that $P(x)$ holds.”
- Then reason to show that $Q(x)$ must hold.
- The word “arbitrary” is important: you do not pick a special $x$. The argument must work for any $x$ satisfying $P(x)$.
2. Non-conditional universal: “For all …, property holds”
Example form:
$$\forall n \in \mathbb{Z}, \; P(n).$$
Often, such a statement can be rewritten as an implication with a trivial hypothesis (for example, “if $n$ is an integer, then $P(n)$”). For a direct proof:
- Start with: “Let $n$ be an arbitrary integer.”
- Use definitions and reasoning.
- Show that $P(n)$ is true.
3. Existence with construction: “There exists … such that …”
Example form:
$$\exists x \in S \text{ such that } P(x).$$
To prove an existence statement directly, you usually:
- Exhibit a specific example $x$ and then
- Directly verify that $P(x)$ holds.
This is sometimes called a constructive proof, but the reasoning from your chosen $x$ to $P(x)$ is direct.
Using definitions in direct proofs
Direct proof relies heavily on rewriting statements using their definitions. Here are a few standard examples of how definitions appear in direct proofs (the full development of these definitions is handled elsewhere; here we only show how they get used):
- “$n$ is even”
Means: there exists an integer $k$ such that $n = 2k$. - “$n$ is odd”
Means: there exists an integer $k$ such that $n = 2k + 1$. - “$a$ divides $b$” (written $a \mid b$)
Means: there exists an integer $k$ such that $b = ak$. - “$x$ is rational”
Means: there exist integers $p$ and $q \neq 0$ such that $x = \dfrac{p}{q}$.
In a direct proof you typically begin by replacing phrases like “$n$ is even” with their formal versions. This allows you to use algebraic manipulation in a precise way.
Flow and style of a direct proof
Beyond the logical structure, direct proofs have a characteristic style.
1. Begin by stating what you’re assuming
Do not silently assume your hypothesis. Write it:
- “Let $n$ be an even integer.”
- “Suppose $a, b \in \mathbb{Z}$ and $a$ is a divisor of $b$.”
- “Let $x$ be a rational number.”
These opening lines clearly mark the starting point of the logical chain.
2. Move in small, justified steps
Each line of your proof should be justifiable from earlier lines using:
- Definitions,
- Algebraic manipulation,
- Known theorems or properties,
- Basic logical rules.
It should be possible, in principle, for someone to check each step. Avoid big leaps like “So clearly…” where several steps are hidden.
3. Aim the argument toward the conclusion
Before you start writing the detailed steps, it helps to look at your desired conclusion and think:
- “What would I need to show to get that?”
- “Can I rewrite the conclusion using definitions so I know what I must produce?”
Then, while proving, you try to shape your work toward that target. A direct proof is a forward-moving chain from assumptions to conclusion, but your planning can (and should) look backward from the conclusion.
4. Finish by matching the exact conclusion
When you reach something equivalent to the statement you want, say so:
- “This is exactly what we needed to show.”
- “Hence, $Q$ holds whenever $P$ holds.”
This shows the reader that the logical chain is complete.
Typical examples of direct proof reasoning
The point of these examples is not the specific results (which are very simple), but to illustrate the direct proof pattern and the use of definitions.
Example 1: Implication between properties
Statement form: If one property holds, another holds.
Example statement:
“If $n$ is an even integer, then $n^2$ is an even integer.”
Outline of a direct proof structure:
- Let $n$ be an even integer.
- By definition of even, there exists an integer $k$ such that $n = 2k$.
- Compute
$$n^2 = (2k)^2 = 4k^2 = 2(2k^2).$$ - Since $k$ is an integer, $k^2$ and $2k^2$ are integers.
- Thus $n^2$ equals $2$ times an integer, which is exactly the definition of “even”.
- Therefore, if $n$ is even, then $n^2$ is even.
Each step follows from the previous ones using standard algebra and the definition of “even”.
Example 2: Universal non-conditional statement
Statement form: A property holds for all integers.
Example statement:
“For all integers $n$, $n^2 + n$ is even.”
Outline of a direct proof structure:
- Let $n$ be an arbitrary integer.
- Consider $n^2 + n$.
- Factor:
$$n^2 + n = n(n + 1).$$ - Among two consecutive integers $n$ and $n+1$, one must be even, so $n(n+1)$ is a multiple of $2$.
- Therefore $n^2 + n$ is even.
The key features are the arbitrary choice of $n$ and the use of a simple property of consecutive integers, together with standard algebra.
Example 3: Existence with a specific example
Statement form: There exists at least one object with a property.
Example statement:
“There exists a rational number whose square is $4$.”
Outline of a direct proof structure:
- Consider the number $x = 2$.
- The number $2$ can be written as $\dfrac{2}{1}$, where $2$ and $1$ are integers and $1 \neq 0$. So $2$ is rational by definition.
- Compute $x^2 = 2^2 = 4$.
- Hence, there exists a rational number $x$ (namely $x = 2$) whose square is $4$.
This proof is direct because we produce a concrete candidate and verify the property using straightforward computation and the definition of “rational”.
When direct proof is (and is not) appropriate
Direct proof works well when:
- You can express the hypothesis using clear algebraic or logical conditions and
- The conclusion is something you can reach by manipulating these conditions via definitions and known facts.
In some situations, however, a direct path from hypothesis to conclusion is awkward or opaque. In those cases, other techniques (covered in the other sections of this chapter on proof techniques) may be more natural, such as:
- Proof by contradiction,
- Proof by contrapositive,
- Mathematical induction.
Direct proof is often the first approach to try. If you find yourself repeatedly stuck trying to go from $P$ to $Q$, it may be a sign that another method is better suited for that problem.
Practical tips for writing your own direct proofs
When you practice direct proof, keep these practical points in mind:
- Translate words into symbols and definitions.
Rewrite informal statements like “$n$ is even” or “$x$ is rational” into precise symbolic forms. This gives you something concrete to manipulate. - Work informally first, then tidy up.
On scratch paper, you might reason in a looser way: “Assume $n$ is even, then $n=2k$, so $n^2=4k^2$… yes, that’s $ times something.”
Once you see the path, rewrite it neatly and clearly as a formal proof. - Keep track of what is arbitrary and what is specific.
For universal statements, your object (like $n$ or $x$) is arbitrary. For existence statements, you must give a specific example. - Use clear logical connectors.
Words like “since”, “therefore”, “hence”, and “so” should correspond to genuine logical steps, not just decoration. Each “therefore” should introduce something that really does follow from what came before. - Match the structure of the statement.
If the statement is of the form “for all $x$, if $P(x)$ then $Q(x)$”, make sure your proof starts with that form: “Let $x$ be arbitrary and suppose $P(x)$ holds.”
A clear match between the statement and your proof structure helps both you and the reader see that the argument is complete.
Direct proof is the foundation upon which many other proof techniques rest. Becoming comfortable with setting up assumptions, using definitions precisely, and chaining together small logical steps will make the more advanced methods much easier to learn and apply.