Lemma — Definition, Meaning & Examples
Lemma
A
helping theorem. A lemma is proven true, just like a theorem, but
is not interesting or important enough to be a theorem. It
is of interest only because it is a stepping stone towards the
proof of a theorem.
See also
Example
Problem: Suppose you want to prove the theorem: 'The sum of two even integers is even.' Before tackling this directly, you first establish a lemma about what it means for a number to be even.
Lemma: State the helper result: If n is an even integer, then n can be written as 2k for some integer k. This fact is straightforward to verify from the definition of 'even,' and we accept it as our lemma.
n is even⟹n=2k for some integer k
Apply the lemma: Let a and b be two even integers. By the lemma, there exist integers j and k such that a = 2j and b = 2k.
a=2j,b=2k
Complete the theorem proof: Add a and b together and factor out the 2.
a+b=2j+2k=2(j+k)
Conclude: Since j + k is an integer, a + b has the form 2 times an integer, so a + b is even. The lemma did the heavy lifting by letting us rewrite each even number in a useful form.
a+b=2(j+k)⟹a+b is even
Answer: The lemma ('every even integer equals 2k for some integer k') was the key stepping stone that made the main theorem ('the sum of two even integers is even') easy to prove.
Frequently Asked Questions
What is the difference between a lemma, a theorem, and a corollary?
All three are statements that have been rigorously proved. A theorem is considered the main, important result. A lemma is a smaller result proved before the theorem to help in its proof. A corollary is a result that follows easily after the theorem has been established. The distinction is about role and perceived importance, not about the level of logical certainty.
Can a lemma become famous or important on its own?
Yes. Some lemmas turn out to be so useful that they become well-known results in their own right. For example, Zorn's Lemma and the Pumping Lemma are famous across mathematics and computer science. They keep the name 'lemma' for historical reasons, even though their significance rivals many theorems.
Lemma vs. Theorem
A lemma and a theorem are both proven statements with equal logical status—neither is 'more true' than the other. The difference is one of purpose and emphasis. A theorem is the headline result that mathematicians care about for its own sake. A lemma is a supporting result whose main job is to simplify or enable the proof of a theorem. In practice, the same statement could be called a lemma in one textbook and a theorem in another, depending on context.
Why It Matters
Lemmas let mathematicians break complex proofs into smaller, manageable pieces. Without them, proofs of major theorems would be extremely long and hard to follow. By isolating a useful fact as a lemma, you can also reuse it across multiple proofs, saving work and improving clarity.
Common Mistakes
Mistake: Thinking a lemma is less certain or less rigorously proved than a theorem.
Correction: A lemma is proved to exactly the same standard of rigor as a theorem. The word 'lemma' indicates its role (a helper result), not a lower degree of certainty.
Mistake: Confusing a lemma with a conjecture or an axiom.
Correction: A conjecture is an unproven claim. An axiom is an assumed starting point accepted without proof. A lemma, by contrast, is fully proved—it just serves a supporting role in a larger argument.
Related Terms
- Theorem — The main result a lemma helps prove
- Corollary — A result that follows easily from a theorem
- Axiom — An assumed truth requiring no proof
- Postulate — Another term for an axiom, common in geometry
- Proof — The logical argument establishing a lemma's truth
- Conjecture — An unproven statement, unlike a lemma