Term Monoid in set theory; A semi group is Monoid If it has identity element with respect to binary operation ∗ .Term Monoid in set theory image.

A non empty set G is said to be monoid if the following condition verify

- G is closed w . r . t binary operation ∗
- G is associative w . r . t binary operation ∗
- G has an identity element w . r . t binary operation ∗

Term Monoid in set theory;

Example :

N = set of natural number N is monoid w . r . t multiplication.

because ∀ ( for all ) a , b ∈ N (N =1, 2 ,3,4…………).

This implies that a × b ∈ N ∈ N ( ∈ means is a part of N).

∀ ( for all ) a , b, c ∈ N (N =1, 2 ,3,4…………).

(This implies that ) ⇒ a × ( b × c ) = (a × b ) × c

Shows that N is associative w .r .t multiplication.

Because 1 ∈ N is an identity element w . r .t multiplication.

But

N = set of natural numbers. N is not Monoid w . r . t multiplication.

because 0 ∉ N ( zero is not a part of natural number).

So N does not have an identity element w . r .t multiplication.

Term Group in set theory :

A monoid is said to be a group if its each element have inverse w . r .t ∗

or

A non empty set G is said to be a group w . r .t binary operation ∗

- G is closed w . r . t binary operation ∗
- G is associative w . r . t binary operation ∗
- G has an identity element w . r . t binary operation ∗
- G has inverse of each element w . r .t binary operation ∗

Term group Example:

Let ℝ – { 0 } be the set of non zero real numbers.

Then

ℝ – { 0 } is a group w . r . t multiplication

Because

- ℝ – { 0 } is closed w . r . t binary operation multiplication.
- ℝ – { 0 } is associative w . r . t binary operation multiplication.
- ℝ – { 0 } has an identity element w . r . t binary operation multiplication.

ℝ – { 0 } has inverse of each element w . r .t binary operation multiplication.

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