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?