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

Question

If Z1 =5i-2j
Z2= 3i+3j
Z3=4i-j,
Show that Z1,Z2 and Z3 are associative, commutative and distributive.

Answer 1

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

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

let's check..

$Z_1+(Z_2+Z_3)\stackrel{?}{=}(Z_1+Z_2)+Z_3$

LHS:

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

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

For commutative property, we must have ...

$Z_i+Z_j=Z_j+Z_i$ for all i and j.

let's check...

$Z_1+Z_2\stackrel{?}{=}Z_2+Z_1$

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

similarly we can show that...

$Z_1+Z_3=Z_3+Z_1=9\hat i-3\hat j\\ Z_3+Z_2=Z_2+Z_3=7\hat i+2\hat j$

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

For distributive property, we must have...

$Z_i.(Z_j+Z_k)=Z_i.Z_j+Z_i.Z_k$ for all i,j and k.

let's check...

$Z_1.(Z_2+Z_3)\stackrel{?}{=}Z_1.Z_2+Z_1.Z_3$

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

Similarly we can show that...

$Z_2.(Z_3+Z_1)=Z_2.Z_3+Z_2.Z_1\\ Z_3.(Z_1+Z_2)=Z_3.Z_1+Z_3.Z_2$

Hence Z_1, Z₂ and Z₃ follow distributive property.