A direct proof is a mathematical proof that uses axioms, definitions, and previously proved theorems without making any further assumptions.^{[1]}
Example: Proof that the sum of two even integers is evenConsider two arbitrary even integers x and y. The definition of an even integer is: An even integer can be written as 2a where a is an integer. We will show that there exists an integer c such that x + y = 2c. If x + y = 2c, then the sum of x and y is even. Using the definition of an even integer, the two integers x and y can be rewritten as x = 2a and y = 2b where a and b are integers. Then the sum x + y can be written as 2a + 2b. Using the distributive property of multiplication, 2a + 2b can be written as 2(a + b). Since a and b are integers, and the set of integers is closed with respect to addition, there exists an integer c = a + b. 2(a + b) can then be rewritten as 2(c) = 2c. Since c is an integer, the expression 2c matches the definition of an even number. QED. | |
Proof in two column form | |
Consider two arbitrary even integers x and y. | Initial assertion |
An even integer can be written as 2a where a is an integer. | Definition of an even integer. |
We will show that there exists an integer c such that x + y = 2c. | Claim |
x and y can be rewritten as x = 2a and y = 2b where a and b are integers. | Apply the definition of an even integer. |
The sum x + y can be written as 2a + 2b. | Substitute 2a for x and 2b for y. |
2a + 2b can be written as 2(a + b). | Apply the distributive property of multiplication. |
There exists an integer c = a + b. | Apply the closure property of integers and addition. |
2(a + b) can then be rewritten as 2(c) = 2c. | Substitute c for a + b. |
QED. | The proof is complete. |
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