1+1=2 Proof

Suzhoumeite

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Jun 15, 2024

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The Proof of 1+1=2: A Mathematical Exploration

The statement "1+1=2" is one of the most fundamental and intuitive mathematical truths. However, have you ever stopped to think about how we can prove this statement to be true? In this article, we'll delve into the world of mathematical proof and explore the different ways to prove that 1+1 indeed equals 2.

Peano Axioms

One way to prove that 1+1=2 is by using the Peano axioms, a set of five axioms that form the foundation of arithmetic. The Peano axioms are:

1. 0 is a natural number.
2. Every natural number has a successor, denoted by S(n).
3. There is no natural number whose successor is 0.
4. If two natural numbers have the same successor, then they are equal.
5. Any property that is possessed by 0 and is possessed by the successor of every natural number that possesses it is possessed by all natural numbers.

Using the Peano axioms, we can define the number 1 as the successor of 0, denoted by S(0). We can then define the number 2 as the successor of 1, denoted by S(S(0)). Therefore, we can prove that 1+1=2 using the following steps:

1. 1 = S(0) (by definition)
2. 1 + 1 = S(0) + S(0) (by substituting 1 with S(0))
3. S(0) + S(0) = S(S(0)) (by the definition of addition in Peano arithmetic)
4. S(S(0)) = 2 (by definition)
5. Therefore, 1 + 1 = 2

Set Theory

Another way to prove that 1+1=2 is by using set theory. In set theory, we can define the number 1 as a set containing a single element, denoted by {a}. We can then define the number 2 as a set containing two elements, denoted by {a, b}. Using the concept of union and intersection of sets, we can prove that 1+1=2 as follows:

1. 1 = {a} (by definition)
2. 1 + 1 = {a} ∪ {a} (by substituting 1 with {a})
3. {a} ∪ {a} = {a, a} (by the definition of union)
4. {a, a} = {a, b} (by renaming the element a to b)
5. {a, b} = 2 (by definition)
6. Therefore, 1 + 1 = 2

Conclusion

In conclusion, the statement "1+1=2" is a fundamental mathematical truth that can be proven using different mathematical frameworks, including Peano axioms and set theory. While the proofs may seem trivial, they demonstrate the importance of rigorous mathematical reasoning and the beauty of mathematical structure.