Answer to Question #150477 in Atomic and Nuclear Physics for Frank Felix

Question #150477
If Z1 =5i-2j
Z2= 3i+3j
Z3=4i-j,
Show that Z1,Z2 and Z3 are associative, commutative and distributive.
1
Expert's answer
2020-12-14T07:25:46-0500

For associative property, elements of set must satisfy following condition :

Zi+(Zj+Zk)=(Zi+Zj)+ZkZ_i+(Z_j+Z_k)=(Z_i+Z_j)+Z_k , where Zi,Zj&ZkZ_i,Z_j \& Z_k are the different elements of a given set.

let's check..

Z1+(Z2+Z3)=?(Z1+Z2)+Z3Z_1+(Z_2+Z_3)\stackrel{?}{=}(Z_1+Z_2)+Z_3

LHS:

LHS:5i^2j^+(3i^+3j^+4i^j^)=12i^RHS:(5i^2j^+3i^+3j^)+4i^j^=12i^LHS:5\hat i-2\hat j+(3\hat i+3\hat j+4\hat i-\hat j)=12\hat i\\ RHS:(5\hat i-2\hat j+3\hat i+3\hat j)+4\hat i-\hat j=12\hat i

Hence, Z2,Z2 and Z3 are associative under addition operation.

For commutative property, we must have ...

Zi+Zj=Zj+ZiZ_i+Z_j=Z_j+Z_i for all i and j.

let's check...

Z1+Z2=?Z2+Z1Z_1+Z_2\stackrel{?}{=}Z_2+Z_1

LHS:(5i^2j^)+(3i^+3j^)=8i^+j^RHS:(3i^+3j^)+(5i^2j^)=8i^+j^LHS: (5\hat i-2\hat j)+(3\hat i + 3\hat j)= 8\hat i+\hat j\\ RHS: (3\hat i + 3\hat j)+(5\hat i-2\hat j)= 8\hat i+\hat j

similarly we can show that...

Z1+Z3=Z3+Z1=9i^3j^Z3+Z2=Z2+Z3=7i^+2j^Z_1+Z_3=Z_3+Z_1=9\hat i-3\hat j\\ Z_3+Z_2=Z_2+Z_3=7\hat i+2\hat j

Hence Z1,Z2 and Z3 are commutative under addition operation.


For distributive property, we must have...

Zi.(Zj+Zk)=Zi.Zj+Zi.ZkZ_i.(Z_j+Z_k)=Z_i.Z_j+Z_i.Z_k for all i,j and k.

let's check...

Z1.(Z2+Z3)=?Z1.Z2+Z1.Z3Z_1.(Z_2+Z_3)\stackrel{?}{=}Z_1.Z_2+Z_1.Z_3

LHS:(5i^2j^).(7i^+2j^)=31RHS:(156)+(20+2)=31LHS: (5\hat i-2\hat j).(7\hat i + 2 \hat j)=31\\ RHS: (15-6)+(20+2)=31

Similarly we can show that...

Z2.(Z3+Z1)=Z2.Z3+Z2.Z1Z3.(Z1+Z2)=Z3.Z1+Z3.Z2Z_2.(Z_3+Z_1)=Z_2.Z_3+Z_2.Z_1\\ Z_3.(Z_1+Z_2)=Z_3.Z_1+Z_3.Z_2

Hence Z1, Z2 and Z3 follow distributive property.

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