Math101 · Independent Ontario learning library
Math101learn.math101.caDirect Proof
A rigorous guide to direct proofs built from definitions, hypotheses, and valid forward implications.
Precise definition
A direct proof of an implication $P\Rightarrow Q$ assumes an arbitrary object satisfying $P$ and derives $Q$ through definitions, established results, and valid algebra or logic. Because the object is arbitrary within the stated domain, the argument covers every permitted case rather than selected examples.
Notation and mathematical language
Write $P(x)$ for the hypothesis and $Q(x)$ for the conclusion. Definitions translate words into usable form: an integer $n$ is even iff $n=2k$ for some $k\in\mathbb Z$, and $a\mid b$ iff $b=ak$ for some integer $k$. Existential witnesses must belong to the required domain.
Conceptual picture
A direct proof is a forward chain: unpack the hypothesis, preserve true statements, and reshape them until the conclusion's definition is visible. Planning often works backward from the conclusion—asking what witness would suffice—but the written proof should make the forward justification clear.
Conditions and key results
The theorem's domain and quantifiers control the proof. Algebraic division requires a nonzero divisor; square roots and inequalities carry domain or sign restrictions. A proof of $P\Rightarrow Q$ need not establish the converse $Q\Rightarrow P$, and using the desired conclusion as a premise is circular.
A reliable strategy
- State an arbitrary object and all hypotheses, including its number system.
- Expand the key definitions in the hypotheses to obtain concrete equations or inequalities.
- Manipulate those statements using justified implications, constructing any witness the conclusion requires.
- Close by matching the exact conclusion and explaining why arbitrariness gives the universal result.
Fully worked example
Interpretation and application
Direct proof is the default language of theorem verification in algebra, number theory, and discrete structures. It also improves program specifications: if input conditions imply an invariant and the invariant plus termination implies the postcondition, the same forward logic establishes correctness.
Common mistakes
Verification and reasonableness
- Annotate each line with the definition or fact that licenses it.
- Test the theorem's boundary cases to detect a missing hypothesis, while remembering tests do not prove it.
- Read the final line against the formal conclusion: the required witness, sign, or set membership should be explicit.
Practice
- Prove the sum of two multiples of $5$ is a multiple of $5$.
- If $n$ is even, why is $n^2$ divisible by $4$?
- Does proving $n$ divisible by $6$ implies $n$ divisible by $3$ prove the converse?
Answers and brief solutions
- Write $a=5r$, $b=5s$; then $a+b=5(r+s)$ and $r+s\in\mathbb Z$.
- $n=2k$ gives $n^2=4k^2$.
- No; for example, $9$ is divisible by $3$ but not by $6$.
Further deduction
Direct inequality proofs require special attention to sign. If $0<a<b$, multiplying by the positive number $ab$ preserves an inequality and yields $a^2<ab<b^2$ in appropriate comparisons; multiplying by a negative quantity would reverse it. Similarly, from $x^2=y^2$ over the reals one may conclude $|x|=|y|$ or $x=\pm y$, not automatically $x=y$. Definitions and domain restrictions determine which algebraic transformations are implications and which are equivalences.
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
If $a=7r$ and $b=7s$, what factor proves $a+b$ is divisible by 7?
- $a+b=7r+7s=7(r+s)$.
- Because $r+s$ is an integer, it is the required witness.
End of lesson
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