Square, that is, quadratic, is a special case of power operation, that is, two squares. That is, the power operation is defined according to the multiplication operation. For "number", square is an operation; The same is true of "setting" and "relationship". The difference between the two is to define the "multiplication" of power.

Multiplication of (1) sets;

It involves the concept of binary group (or ordered pair), that is, the form such as

For two sets A and B, choose one element A and B from A and B respectively to form a binary group.

With multiplication, you can naturally define the power: a? = A×A .

For example, all points on the number axis correspond to a real number; Every point in the plane coordinate system corresponds to a coordinate. In fact, the coordinate is a binary group (x, y): x comes from the horizontal axis; Y comes from the vertical axis. The horizontal axis and the vertical axis both correspond to the real number set R. Therefore, the plane coordinate system can be expressed as: R×R, that is, r? .

(2) the relationship of "multiplication":

First of all, the relationship itself is a set. Therefore, any relation can be multiplied by a set, that is, the Cartesian product, and its rules are as described in (1). If this is the square of your relationship, then you don't have to look behind.

However, the relationship is a special set, so a special "multiplication" operation-the combination of relationships is produced.

Similar to (1), we must first define "multiplication", that is, "combination of relations"; Then, the definition of square is very simple: the square of a relationship is related to its own composition.

As for the definition of compound, I don't need to tell you here. You can read this book by yourself. Just to remind you:

In view of the relationship itself is a set, if the same symbol is used to represent the power of the relationship, it will inevitably lead to ambiguity, so in mathematics, the unique "multiplication" and "power" of the relationship-the combination of the relationship-have stipulated new symbols:

R o S: the combination of relations r and s;

S O S = s (2): the quadratic compound of relation S to itself, that is, the square of S; ("Exponent" 2 needs to be enclosed in brackets)

On the contrary, the cartesian product of s as a set is expressed as:

S × S = S ^ 2 = S？ .

In fact, we no longer call the combination of relations "multiplication" or "power", but "combination of R and S" or "N-fold combination of S to itself". It is only because these two operations are very similar to multiplication and power that they are called in some informal situations. The cartesian product of a set can be called "multiplication between sets", and the theory of "power" of a set is also logical. Therefore, the relationship only appears as a set, which is the real "square". )