For many people, the question that seems incredibly simple: “Why does 1 + 1 = 2?” is one of the hardest questions to answer. Why? Because it seems almost obvious. You have 1 apple, and then someone gives you another apple, so you have 2 apples; it appears to be that way naturally.
Proving that 1 + 1 does not equal 2
However, from the perspective of modern mathematics, proving that “1 + 1 = 2” is redundant because it does not hold any further significance. In fact, one could even prove that “1 + 1” does not equal 2.
Let me present to you a construction where “1 + 1” will no longer equal 2 but some arbitrary value according to the principles of mathematics.
First, we need to define a few basic concepts:
1. Set
This is a fundamental concept in mathematics. Therefore, we do not have an answer to “What is a set?” Instead, when we talk about a set, we refer to the objects within it, which we call elements. Thus, we can refer to a set by the properties of its elements.
For example: “The set of natural numbers” consists of the elements 0, 1, 2, 3,…
“The set of vehicles on the road” includes elements like cars, motorcycles, bicycles…
Sets are usually denoted by capital letters, such as set A, set B, set of natural numbers N,…
In this article, we will examine an operation on sets called Cartesian product. For two sets A and B, the Cartesian product of A and B is denoted as AxB, which is a set comprising elements in the form (x; y), where x is an element of A, and y is an element of B (following the specific order).
2. Mapping
For two sets X and Y, a correspondence “each element x of X with a unique element y of Y” is called a mapping.
In this definition, we must note that if x belongs to X, there must be, and only one, element y belonging to Y corresponding to x. If there is an x without a corresponding y or if there are two elements in Y corresponding to x, it is not called a mapping.
Mappings are denoted as f from X to Y, and the image of element x from X is denoted as f(x).
3. Constructing the problem model
After having the two concepts above, let’s build a model for the problem 1 + 1 does not equal 2:
Let the set of natural numbers N and the set of types of fruits, denoted as T. The Cartesian product of set N and N is NxN, consisting of elements in the form (a; b) (which we call the pair (a; b)), where a and b are natural numbers.
Consider the mapping f from the set NxN to set T, where each pair of numbers (a; b) corresponds to the name of a certain type of fruit, denoted as f(a; b). We denote f(a; b) = a + b (note that a + b here is just a notation).
When considering the pair (1; 1), it corresponds to a name of some fruit in set T (it must exist according to the definition of mapping), let’s assume it is “Orange.” Then we get
f(1; 1) = “Orange”, or in other words, we have “1 + 1 = Orange” (since f(1; 1) = 1 + 1).
4. Conclusion
From the model above, we have reached the conclusion that 1 + 1 is no longer 2, but it can be anything we want. Furthermore, from this model, we also have an answer to “Why does 1 + 1 = 2?” That is: this is simply a convention of mathematical operations established by humans, thus humans can completely change it (for instance, instead of using the “+” symbol, one could use the “-” symbol, and then we would have “1 – 1 = 2”, which essentially changes nothing, only the notation is altered).
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