# Set Theory Symbols

A set can be described as a collection of objects. The objects of a particular set are termed to be its elements. The creation history of set theory is unique compared to other areas of mathematics. The main reason the set theory was created was to discuss collections of objects representing a particular set. To describe some of the most complex and mathematical structures, set theory can be utilised and is considered a very important tool.
The following are some of the most common examples of sets:

• Collection of greatest football players.
• Collection of children born after 1995.
• The collection of positive numbers.
• Collection of supercars.
• Collections of silk dresses.

In the above examples, the collection of the greatest football players is not well-defined. So it is better to focus only on defined sets in order to obtain accurate results.
Below mentioned are the three ways in which one can define sets:

• Descriptive form
• Listing form
• Set builder form

For instance, the following is the representation of a set of odd natural numbers that lie between the interval of 2 and 14 in all three forms, such as descriptive, roster (listing) and set builder form:

Descriptive form: Example - The collection of all the odd counting numbers from 2 to 14.

Roster (listing) form: Example - {3, 5, 7, 9, 11, 13}.

### Set builder form: Example – {x | x is a natural number, whereas x is odd, and 2 < x < 12}. These are some of the requisites that need to be followed:

• Curly braces should be used to represent sets.
• Commas should be used to split the elements of the set from each other.
• The variable contained in the set-builder notation does not necessarily have to be x.
• The ellipses (….) given above should be used to indicate the continuation of patterns that have been established before the ellipses, {1, 2, 3,….,99, 100}
• The symbol ‘|’ mentioned above is called “such that”.

### Below mentioned are some of the common symbols used while defining sets:

• ∈ Denotes ‘an element of’. For example: Is 2∈ {2, 4, 6}? The answer is yes.
• N indicates Natural numbers, {1, 2, 3, 4, 5,..}. They can also be called counting numbers.
• W states Whole numbers, {0, 1, 2, 3, 4,..}.
• I denotes Integers, {-3, -2, -1, 0, 1, 2, 3,..}.
• Q indicates Rational numbers, {p/q | p, q ∈ I, q 6 = 0}.
• R demonstrates Real numbers, {x | x ∈ R, x is a number which can be written as decimal}.
• Irrational numbers, {x | x is a real number, but it cannot be expressed as a quotient of integers}.
• ∅ Indicates an Empty set, {}. It means it contains nothing.
• U represents the Universal set. It is the set of all objects that are currently under discussion.
• U This is nothing but a set union. It means the set of elements belong to either of the sets. Example: X⋃ Y.
• ⋂ This indicates intersection. It means the set of elements that are common in both sets. Example: X⋂ Y.
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