**What is Groupoid in Mathematics**?

A groupoid is a non empty set on which a binary operation (steric)* is defined.**in simple words:**A closed set with respect to an operation (steric) is said to be

**groupoid**.

**What is Semi Group Explain the Satisfy Condition?**

A non empty set **S **is said to be** semi group** if it satisfy the following:

**⇒**It it closed w.r.t an operation (steric).**⇒**The operation (steric) is associative it means it holds associativity.

**What is Monoid ****Explain the Satisfy Condition** ?

**Explain the Satisfy Condition**?

A semi group or set S is called **Monoid** if it Satisfy the following:

**⇒**it is closed w.r.t some operation (steric).**⇒**The operation (steric) holds associativity.**⇒**It has an identity element.

**in simple words:**A semi group which have identity element is used to be said

**Monoid**.

**What is Group in Math?**

A Monoid which have inverse of each of its elements under the operation (steric) is used to be said group.

in simple manners, a set is said to be a group if it Satisfy the following:

**⇒**G is closed w.r.t some operation (steric).**⇒**The operation (steric) holds associativity.**⇒**It has an identity element.**⇒**every element of G has its inverse in G

**How a group become abelian group?**

**An additional condition is:⇒ a × b = b × a**

it means that if it holds commutativity (

**commutative property**) then G is an

**abelian**group.

**Natural Numbers w.r.t to Addition **

**N={1,2,3,…}**

for Addition:

according to first property:

N should be closed i.e., 1+1=2 (2 also in N) It should hold associativity i.e.,

3+(5+8)= (3+5)+8

16=16

Identity exists

as identity element is 0 that doesn’t exist in N

Conclusion

N is a semi_group w.r.t Addition.

` Whole numbers w.r.t to Addition`

W={0,1,2,3,…}

i W should be closed i.e.,

3+4=7 (7 is also in W)

ii It should hold associativity i.e.,0+(3+4)=(0+3)+4

7=7

iii additive identity is 0 that also exists in W

7=7

Conclusion

W is a Monoid w.r.t to addition.

` Even numbers w.r.t to Addition`

E={2,4,6,…}

i E should be closed i.e.,

4+10=14 (14 also exists in E).

ii It should hold associativity i.e.,

4+(8+10)=(4+8)+10

22=22

iii additive identity is 0 that also exists in E because 0 is an even number.

iv Inverse does not exists in the E.

conclusion

E is Monoid w.r.t Addition.

` Odd numbers w.r.t to Addition`

O={1,3,5…}

i O should be closed i.e.,

3+5=8 (8 doesn’t exists in O).

Conclusion

set of odd Numbers is not a group w.r.t addition.

` Prime numbers w.r.t to Addition`

p={3,5,7,11,…}

i P should be closed i.e.,

3+7=10 (10 doesn’t exist in P).

ii Associative law holds.

Identity element is 0 that doesn’t exist in P.

Conclusion

set of prime numbers is not a group w.r.t addition.

` Composite numbers w.r.t to Addition`

c={4,6,8…}

i C should be closed i.e.,

8+9=17 (17 doesn’t exist in P)

Conclusion

it is also not a group w.r.t addition.

` Rational Numbers w.r.t to Addition`

R={0,±1,±2,…}

i R should be closed i.e.,

1+1=2 & -1-1=-2 (closure property exists)

ii It should hold associativity i.e.,

1+(2+3)=(1+2)+3

6=6

iii additive identity that is*0* , it also exists in R.

iv inverse is also available in R i.e., additive inverse of 1 is -1.

Conclusion

it is a group w.r.t addition w.r.t Addition.

` Z^+ w.r.t to Addition`

z={±1,±2,±3,…}

i z^+ should be closed i.e.,

1+2=3 (exists in set z^+)

ii It holds associativity.

iii Additive identity that is 0 , doesn’t exist in set z^+.

conclusion

it is a semi_group w.r.t addition w.r.t Addition.

` Z^- w.r.t to Addition`

z={-1,-2,-3,…}

i z^- should be closed i.e.,

-1+(-4)=-5 (exists in the set)

ii associativity holds in the set.

iii Additive identity that is 0, doesn’t exist in the set.

Conclusion

set of Z^- integers is semi_group w.r.t Addition.