What are the properties of a set of all real numbers?

Answer 1

The real number system is a mathematical field.

To start with, the Real number system is a Group. This means that it is a set of elements (numbers) with a binary operation (addition) that combines any two elements in the set to form a third element which is also in the set. The Group satisfies four axioms: closure, associativity, identity and invertibility.

Closure: For all x and y in the set, x+y is also in the set.

Associativity: For all x, y and z in the set, (x+y)+z = x+(y+z)

Identity: there exists an element, normally denoted by 0, such that for any element x in the set, 0+x = x = x+0.

Invertibility: For every element x in the set, there is an element y in the set such that x+y = 0 = y+x. y is usually written as -x.

In addition, it is a Ring. A ring is an Abelian group (that is, addition is commutative) and it has a second binary operation (multiplication) that is defined on its elements. This second operation is distributive over the first.

Abelian (commutative): For every x, y in the set, x+y = y+x.

Distributivity of multiplication over addition: For any x, y and z in the set, x*(y+z) = x*y + x*z.

And finally, a Field is a Ring over which division - by non-zero numbers - is defined.

The four axioms of a Group are satisfied for the second operation. The multiplicative identity is denoted by 1, the multiplicative inverse of an element x is denoted by 1/x or x^-1.