Whenever we read a book, a research paper, or solving homework problems, we come across terms like theorem, lemma, axiom, and more. Each has a specific meaning and role in mathematical writing.

In this post, we’ll walk through these terms using a simple example: even and odd numbers.

Axiom

Definition: A basic assumption accepted without proof. Axioms serve as the foundation for reasoning in a mathematical system.

Example:
“Every integer is either even or odd.”
This is a basic truth we assume in number theory—no need to prove it.

Definition

Definition: A precise explanation of a mathematical concept.

Example:
“An integer ( n ) is even if there exists an integer ( k ) such that ( n = 2k ).”
This tells us exactly what “even” means and sets the groundwork for any future statements involving even numbers.

Theorem

Definition: A major mathematical statement that must be proven using axioms, definitions, and previously proven results.

Example:
Theorem: The sum of two even integers is even.
This is a core fact that can be proven logically from the definition of even numbers.

Lemma

Definition: A minor or intermediate result used to prove a more important theorem.

Example:
Lemma: If ( a = 2k ) and ( b = 2m ), then ( a + b = 2(k + m) ).
This helps us prove the theorem above, but is typically not interesting on its own.

Proof

Definition: A logical argument that demonstrates a statement is true, based on accepted premises.

Example (Proof of Theorem):
Let ( a = 2k ), ( b = 2m ). Then
[ a + b = 2k + 2m = 2(k + m) ]
Since ( k + m ) is an integer, ( a + b ) is divisible by 2, so it is even. ∎

Corollary

Definition: A statement that follows directly and easily from a theorem.

Example:
Corollary: The sum of any number of even integers is even.
This follows by applying the theorem repeatedly.

Proposition

Definition: A proven statement like a theorem, but generally considered less central or important.

Example:
Proposition: If ( n ) is even, then ( n^2 ) is even.
While this is useful, it’s not a deep result—but it still requires proof.

Claim

Definition: A statement asserted (and usually proven) as a step within a larger proof.

Example (within a longer proof):
Claim: ( k + m ) is an integer.
This might be a small step used to show that ( 2(k + m) ) is even.

Conjecture

Definition: A mathematical statement believed to be true but not yet proven or disproven.

Example:
Goldbach’s Conjecture: Every even integer greater than 2 is the sum of two primes.
Despite much evidence, this remains unproven.

Common Types of Mathematical Proofs

Mathematical proofs are not all the same—different statements require different strategies. Below are five foundational proof techniques with clear, worked examples to help you understand how and when each method is used.

Proof by Construction

What it is: You prove that something exists by explicitly creating or exhibiting an example of it.

Example:
Claim: There exists an even prime number.
Proof:
Consider the number ( 2 ).
- ( 2 ) is divisible by ( 2 ), so it is even.
- The only positive divisors of ( 2 ) are ( 1 ) and ( 2 ), so it is prime.
Thus, an even prime exists. ∎

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Proof by Induction

What it is: A method for proving statements about natural numbers. You prove the base case, then assume the statement for ( n = k ) (the inductive hypothesis) and use it to prove the statement for ( n = k + 1 ).

Example:
Theorem:
For all ( n ),
[ 1 + 2 + + n = ]

Proof:
Base case: ( n = 1 ).
LHS: ( 1 ), RHS: ( = 1 ). ✓

Inductive hypothesis: Assume the formula holds for ( n = k ), i.e.,
[ 1 + 2 + + k = ]

Inductive step: Show it holds for ( n = k+1 ).
Add ( (k + 1) ) to both sides:
[ 1 + 2 + + k + (k + 1) = + (k + 1) ]

Factor RHS:
[ = ]

This matches the formula with ( n = k+1 ). ∎

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Proof by Contradiction

What it is: You assume the negation of the statement you want to prove, then show that this assumption leads to a logical contradiction. This implies the original statement must be true.

Example:
Claim: ( ) is irrational.

Proof:
Assume the opposite: that ( = ) where ( a ) and ( b ) are integers with no common factors.

Then: [ 2 = a^2 = 2b^2 ]

So ( a^2 ) is even ⇒ ( a ) must be even (since square of an odd is odd).
Let ( a = 2k ). Then: [ a^2 = (2k)^2 = 4k^2 = 2b^2 b^2 = 2k^2 b ) is also even.

But now both ( a ) and ( b ) are even ⇒ they share a factor of 2, contradicting our assumption that the fraction is in lowest terms.
Therefore, ( ) is irrational. ∎

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Proof by Contrapositive

What it is: Instead of proving “If ( P ), then ( Q )”, you prove the logically equivalent statement: “If not ( Q ), then not ( P )”.

Example:
Claim: If ( n^2 ) is even, then ( n ) is even.

Proof (contrapositive):
We prove: If ( n ) is odd, then ( n^2 ) is odd.

Let ( n = 2k + 1 ) for some integer ( k ).
Then: [ n^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1 ]

This is of the form ( 2m + 1 ), so it is odd.
Therefore, if ( n ) is odd, ( n^2 ) is odd ⇒ contrapositive is true ⇒ original statement is true. ∎

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Proof by Exhaustion (Case Analysis)

What it is: You divide a problem into a finite number of cases and prove the statement holds in each one.

Example:
Claim: For any integer ( n ), ( n^2 ) or ( 1 ).

Proof:
There are four residue classes mod 4: ( 0, 1, 2, 3 ). Consider all cases:

• ( n n^2 )
• ( n n^2 )
• ( n n^2 )
• ( n n^2 )

In all cases, ( n^2 ) or ( 1 ). ∎

@article{gopal2024commontermsusedinmathematicalproofs,
  title   = {Common Terms used in mathematical proofs},
  author  = {Senthilkumar Gopal},
  journal = {sengopal.me},
  year    = {2024},
  month   = {May},
  url     = {https://sengopal.me/posts/common-terms-used-in-mathematical-proofs}
}