# Set theory proof 2

1. Apr 2, 2016

### Jairo Rojas

1. The problem statement, all variables and given/known data
. Let A be a set and {B1, B2, B3} a partition of A. Assume {C11, C12} is a partition of B1, {C21, C22} is a partition of B2 and {C31, C32} is a partition of B3. Prove that {C11, C12, C21, C22, C31, C32} is a partition of A.

2. Relevant equations

3. The attempt at a solution
I know this problem looks a little bit intuitive but I just want to make sure I make sense
proof

so {C11, C12, C21, C22, C31, C32} is a partition of A because A is divided into 3 disjoint subsets which make a partition of A, B1,B2,B3. And then each subset of these 3 disjoint subsets is further divided into two disjoint subsets which make a partition of the parent subset. So all elements in the parent subsets will make a partition of A because every element in each parent subset are disjoint among the elements of every other parent subset.

2. Apr 2, 2016

### Orodruin

Staff Emeritus
Your idea is correct, but could be put in a bit more formal way, i.e., take an element in A and show that it must belong to one and only one of the sets Cij.

3. Apr 2, 2016

### Jairo Rojas

ok this is what I did
proof
x∈A
so x∈(Just one of B1,B2,B3 because disjoint by definition)
case 1 x∈A
so x∈ Just one of C11,C12 because disjoint by definition

case 2 x∈B
so x∈ Just one of C21,C22 because disjoint by definition

case 3 x∈C
so x∈ Just one of C31,C32 because disjoint by definition

which implies that x∈(Just one of C11,C12,C21,C22,C31,C32)

4. Apr 3, 2016

### Orodruin

Staff Emeritus
I would also specify that x being in B1 rules out it being in any other C

5. Apr 3, 2016

### HallsofIvy

Staff Emeritus
You mean "x∈B1"

You mean "x∈B2"

You mean "x∈B3"

Last edited by a moderator: Apr 3, 2016
6. Apr 3, 2016

### Jairo Rojas

yes, didn't notice that. Is the logic right?. How does this show that these elements make a partition of A?