Math 2.0

Group theory and generalizations Example

The natural numbers under addition are not a group

Builds on Natural numbersInverse element

Example

Let ℕ={0,1,2,… }\mathbb{N} = \{0, 1, 2, \dots\} be the natural numbers with addition. Addition is associative, and 00 is an identity element. The element 11 has no inverse, because 1+m≥11 + m \ge 1 for each m∈ℕm \in \mathbb{N}, so 1+m≠01 + m \ne 0. Thus (ℕ,+)(\mathbb{N}, +) is not a group. It is a commutative monoid.

Remarks

If we add the negative integers, we get the group (ℤ,+)(\mathbb{Z}, +).

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Binary operation, definition, no evidence levelBinary operationdefinitionNatural numbers, object, no evidence levelNatural numbersobjectIdentity element, definition, no evidence levelIdentity elementdefinitionInverse element, definition, no evidence levelInverse elementdefinitionThe natural numbers under addition are not a group, example, no evidence levelThe natural numbers u…example
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  1. Binary operation definition
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  3. Identity element definition
  4. Inverse element definition
  5. The natural numbers under addition are not a group example

Further reading

  1. Web https://en.wikipedia.org/wiki/Natural_number, Addition
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Not an instance ofGroup (note: the element 1 has no inverse.)
UsesNatural numbers, Inverse element

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