Prove by Contradiction: For all integers x greater than 11

Yes, for odd numbers, we can use the fact that any odd number can be written as 2k+1, where k is an integer. Since 2 is prime, 2k is composite, and 1 is also composite. Therefore, any odd number can also be written as the sum of two composite numbers. In summary, to prove by contradiction, we assume the negation of the statement and then show that a contradiction arises. In this case, we can show that for all integers x greater than 11, x can be written as the sum of two composite numbers, both for even and odd numbers. Therefore, our original statement holds true.
  • #1
Animuo
13
0

Homework Statement


Prove by Contradiction: For all integers x greater than 11, x equals the sum of two composite numbers.


Homework Equations


A composite number is any number that isn't prime
To prove by contradiction implies that if you use a statement's as a negation, a contradiction arises

The Attempt at a Solution


The negation of the original statement is:
There exists an integer x such that if x > 11, then x does not equal the sum of two composite numbers.

I'm really stuck on this one, I tried substituting in values using the quotient remainder theorem (a number can be represented as 2r, or 2r + 1.. or alternatively 3r, 3r + 1, or 3x + 2, but I wasn't getting anywhere with it). Some direction would be appreciated!
 
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  • #2
Given that x is not the sum of two composite numbers, what can you say about things like

x-4
x-6
x-8
x-9
 
  • #3
All even numbers are composite (besides 2!), and since 4 is composite as well, any even number greater than 4 can be written as the sum of two composite numbers. This leaves only odd ones left.

Can you apply a similar logic for odd numbers?
 

Related to Prove by Contradiction: For all integers x greater than 11

1. What is the concept of "Prove by Contradiction"?

Prove by Contradiction is a method of mathematical proof where we assume the opposite of what we are trying to prove, and then show that this assumption leads to a contradiction. This contradiction proves that the original statement is true.

2. How does "Prove by Contradiction" apply to the statement "For all integers x greater than 11"?

In this statement, we would assume the opposite, which is "There exists an integer x greater than 11 that does not satisfy the statement." We would then show that this assumption leads to a contradiction, thus proving that the original statement is true for all integers x greater than 11.

3. Why is "Prove by Contradiction" a useful proof technique?

Prove by Contradiction allows us to prove a statement by showing that its opposite leads to a contradiction. This can be especially useful when direct proof or proof by induction is not possible or would be too complicated. It also helps to strengthen our understanding of a concept by examining its opposite.

4. Are there any limitations to using "Prove by Contradiction"?

One limitation is that it can be difficult to find the contradiction when working with more complex statements. It also does not provide any insight into why the statement is true, only that it cannot be false. Additionally, it may not be appropriate for all types of mathematical proofs.

5. Can "Prove by Contradiction" be applied to any type of statement?

Prove by Contradiction is most commonly used for proving universal statements, such as "For all integers x greater than 11". However, it can also be used for existential statements, as long as the contradiction can be found within the statement itself. It is not applicable for proving statements that are not logically equivalent to their negation.

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