Consider the following conjecture: the sum of two even numbers will be even.
A.) Prove the conjecture
To be even, you have to be divisible by 2.
x/2 + y/2 =( x + y)/2
even numbers have a difference of 2.
n + n + 2 = 2n +2 which is divisible by 2
or something like
n+2 + n+4 = 2n+6 again divisible by 2.
To prove the conjecture, we need to use the definition of even numbers and perform a mathematical proof.
Definition: An even number is an integer that can be divided evenly by 2.
Let's consider two even numbers, x and y.
By definition, we can write x as 2a, where 'a' is an integer, and also write y as 2b, where 'b' is an integer.
Now let's calculate the sum of x and y:
x + y = 2a + 2b
We can factor out 2 from both terms:
x + y = 2(a + b)
Since a and b are integers, let's define a new integer k = a + b.
So, we can rewrite the sum as:
x + y = 2k
By definition, 2k is divisible evenly by 2. Therefore, the sum of two even numbers, x and y, is even.
Hence, the conjecture has been proven.
Note: In a mathematical proof, it is important to start from the definitions, show the logical steps, and provide a clear conclusion in order to validate a conjecture.