AUB = {1,2,3}, then how many pairs of sets a and B satisfy the condition

AUB = {1,2,3}, then how many pairs of sets a and B satisfy the condition

The basic subjects of union knowledge are listed one by one if they are not proficient
A is &;, B {1,2,3}
A is {1}, B {2,3} or {1,2,3} - two groups
A is {2}, B {1,3} or {1,2,3} - two groups
A is {3}, B {1,2} or {1,2,3} - two groups
A is {1,2}, B {3} or {1,3} or {2,3} or {1,2,3} - 4 groups
The detailed enumeration of group A is {2,3} - 4 and group A is {1,3} - 4 is omitted here
A is {1,2,3}, B {1} or {2} or {3} or {1,2} or {1,3} or {2,3} or {1,2,3} or # 8709; - 8 groups
If you are proficient, you can directly calculate 1 + 2 * 3 + 4 * 3 + 1 + 3 + 3 + 1 = 27, there are 27 pairs in total

How many sets a and B satisfy AUB = {1,2}?
Answer with the principle of classified addition and multiplication
Answer in detail

A = {1}, {2}, {1,2}, empty sets
B = {1}, {2}, {1,2}, empty sets
According to the principle of classified addition and counting
So we use the exhaustion method
There is a = {1}; b = {2}
A = {2}; b = {1}
A = {1,2}; b = {1}, {2}, {1,2}, four kinds of empty sets
A = {1}, {2}, {1,2}, empty set; b = {1,2}
That adds up to 10

If the set a = {1,2}, then the number of sets B satisfying the condition AUB = {1,2} is 4,..., why

A=A∪B
So B is a subset of A
A is two elements
So the subset of a has 2 to the power of 2
So B has four
If a has n elements
Then there are two powers of A