The set of all real numbers except 100, denoted as R-100, is a fascinating subset of the real numbers with unique properties and applications. This discussion will delve into its mathematical definition, explore its relationship with other number sets, and uncover its significance in various branches of mathematics.

R-100 is defined as the set of all real numbers excluding 100. It possesses several intriguing properties, including closure under arithmetic operations and an uncountable cardinality. These properties make it distinct from other number sets, such as the integers, rational numbers, and irrational numbers.

The Set of All Real Numbers Except 100

The set of all real numbers except 100 is denoted by R\100. It is the set of all real numbers except the number 100. This set is often used in mathematics to study the properties of the real numbers and their applications.

Definition and Properties of the Set of All Real Numbers Except 100

The set of all real numbers except 100 can be defined as follows:

R\100 = x ∈ R | x ≠ 100

where R is the set of all real numbers.

The set R\100 has several properties, including:

• It is a subset of the real numbers.
• It is an infinite set.
• It is closed under addition, subtraction, multiplication, and division (except division by 0).
• Its cardinality is equal to the cardinality of the real numbers.

Relationship with Other Number Sets, The set of all real numbers except 100

The set R\100 is related to other number sets as follows:

• It is a proper subset of the set of all real numbers.
• It is a disjoint set from the set of integers.
• It is a proper superset of the set of rational numbers.
• It is a proper subset of the set of irrational numbers.

Applications in Mathematics

The set R\100 has several applications in mathematics, including:

• It is used in calculus to study the limits of functions.
• It is used in algebra to study the properties of polynomials.
• It is used in geometry to study the properties of circles.

Visualization and Representation

The set R\100 can be visualized as a number line with a point at 100 removed.

Another way to represent the set R\100 is using the interval notation:

R\100 = (-∞, 100) ∪ (100, ∞)

Examples and Counter-Examples

Examples of numbers that belong to the set R\100 include:

• 0
• π
• -1
• 101

Examples of numbers that do not belong to the set R\100 include:

• 100
• 200
• -100

Essential Questionnaire: The Set Of All Real Numbers Except 100

What is the cardinality of R-100?

The cardinality of R-100 is equal to the cardinality of the real numbers, which is uncountable.

Is R-100 closed under multiplication?

Yes, R-100 is closed under multiplication. The product of any two numbers in R-100 is also in R-100.

What is an example of a number in R-100?

Any real number other than 100 is an example of a number in R-100. For instance, 5, -2.7, and √2 are all in R-100.