Direct proof

A direct proof is the most straightforward proof method in mathematics. Here one starts from the given assumptions and reasons step by step towards the conclusion to be shown.

Each step is based on known rules, definitions, or previously proven results.

 

The method is particularly well suited when working with statements of the type "if ... then ...".

One begins by assuming that the condition is fulfilled, and then shows logically that the conclusion must follow. In this way, a direct proof resembles a chain of arguments, where each link follows naturally from the previous one.

 

Procedure

When making a direct proof, one can often structure the work in three steps:

 

1. Assume the premises are true.
2. Use definitions, rules, and known theorems to derive new results.
3. Continue until the stated conclusion is shown.

 

 

Example 1

We want to prove that the sum of two odd numbers is always even. Let the numbers be written in the form \( \large 2a+1 \) and \( \large 2b+1 \). Then we get:

 

$$ \large (2a+1) + (2b+1) = 2a + 2b + 2 = 2(a+b+1) $$

 

The result can be written as 2 times an integer, that is, an even number. Thus, the statement is proven directly.

 

 

Example 2

We want to prove that the product of two even numbers is always divisible by 4. Write the numbers as \( \large 2m \) and \( \large 2n \). Then:

 

$$ \large (2m)\cdot(2n) = 4mn $$

 

Since the expression can be written as 4 times an integer, it is always divisible by 4. Thus, the statement is proven by direct proof.

 

 

Example 3

We want to prove that if two numbers are both divisible by 3, then their sum is also divisible by 3. Let the numbers be \( \large 3x \) and \( \large 3y \). Then we get:

 

$$ \large 3x + 3y = 3(x+y) $$

 

The sum is a multiple of 3 and therefore divisible by 3. Thus, the statement is proven directly.

 

 

Direct proofs provide a clear and intuitive understanding of why a statement is true. The method is also the foundation for many other proof techniques, which build on the same idea: that logical rules can connect the assumptions with the conclusion.