Final answer:
The given statement "Proof by contradiction: Every integer greater than 11 is a sum of two composite numbers." is false, thus the correct option is B.
Step-by-step explanation:
To prove the statement, 'Every integer greater than 11 is a sum of two composite numbers,' we will use the proof by contradiction method. This method assumes the opposite of the statement and then shows that it leads to a contradiction, thereby proving the original statement to be true.
Assuming the opposite of the given statement, let us say that there exists an integer greater than 11 that is not a sum of two composite numbers. In other words, this integer can only be represented as a sum of a prime number and a composite number. Let us denote this integer by n.
Now, according to the fundamental theorem of arithmetic, every integer greater than 1 can be expressed as a unique product of prime numbers. Therefore, n can be expressed as n = p₁p₂, where p₁ and p₂ are prime numbers. Since n is greater than 11, both p₁ and p₂ must be greater than 1.
Next, we will consider two cases:
Case 1: p₁ is even
If p₁ is even, then it can be expressed as p₁ = 2k, where k is a positive integer. Therefore, n can be written as n = 2kp₂. This means that n is a multiple of 2 and p₂, and hence it is a composite number. But this contradicts our assumption that n cannot be expressed as a sum of two composite numbers. Therefore, our assumption was false and the original statement must be true.
Case 2: p₁ is odd
If p₁ is odd, then it must be greater than 2, since 2 is the only even prime number. This means that p₁ can be expressed as p₁ = 2k + 1, where k is a positive integer. Therefore, n can be written as n = (2k + 1)p₂. Again, this shows that n is a multiple of 2 and p₂, and hence it is a composite number. This contradicts our assumption and proves that the original statement is true.
Hence, we can conclude that every integer greater than 11 is indeed a sum of two composite numbers.
In this proof, we have used the fundamental theorem of arithmetic, which states that every integer greater than 1 can be expressed as a unique product of prime numbers. This theorem is crucial in understanding the composition of numbers and is widely used in number theory. By breaking down an integer into its prime factors, we can easily determine whether it is a sum of two composite numbers or not.
In conclusion, the proof by contradiction method has helped us to prove the given statement to be true. Through logical reasoning and the use of fundamental mathematical concepts, we have arrived at the final answer that every integer greater than 11 is indeed a sum of two composite numbers, thus the correct option is B.