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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.

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

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By definition, it is the set of all real numbers!

Are disjoint and complementary subsets of the set of real numbers.

The set of Natural Numbers is the set of 'counting numbers' {1,2,3,4,....}. All of them are also real numbers.

It's a set with an infinite quantity of elements, like the set of all real numbers, or the set of all real numbers except zero, etc.

Not by itself. A mathematical operation has properties in the context of a set over which it is defined. It is possible to have a set over which properties are not valid.Having said that, the set of rational numbers is closed under subtraction, as is the set of real numbers or complex numbers.Multiplication is distributive over subtraction.

Related questions

the set of real numbers

real numbers

By definition, it is the set of all real numbers!

The set of all real numbers (R) is the set of all rational and irrational numbers. The set R has no restrictions in its domain and so includes (-∞, ∞).

Are disjoint and complementary subsets of the set of real numbers.

The set of all real numbers (R) is the set of all rational and Irrational Numbers. The set R has no restrictions in its domain and so includes (-∞, ∞).

The set of Natural Numbers is the set of 'counting numbers' {1,2,3,4,....}. All of them are also real numbers.

It's a set with an infinite quantity of elements, like the set of all real numbers, or the set of all real numbers except zero, etc.

There are rational numbers and irrational numbers. Real numbers are DEFINED as the union of the set of all rational numbers and the set of all irrational numbers. Consequently, all rationals, by definition, must be real numbers.

Not by itself. A mathematical operation has properties in the context of a set over which it is defined. It is possible to have a set over which properties are not valid.Having said that, the set of rational numbers is closed under subtraction, as is the set of real numbers or complex numbers.Multiplication is distributive over subtraction.

real numbers

Yes - in fact the set of all even numbers is a subset of the set of all integers, which is, in turn, a subset of the set of all real numbers.