Proof of Partition of a Set with Nested Partitions | Set Theory Proof 2

[Jairo Rojas]

Homework Statement

. 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.

Homework Equations

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.

 

[Orodruin]

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.

 

[Jairo Rojas]

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.

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)

 

[Orodruin]

Just one of C11,C12 because disjoint by definition

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

 

[HallsofIvy]

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

You mean "x∈B1"

so x∈ Just one of C11,C12 because disjoint by definition

case 2 x∈B

You mean "x∈B2"

so x∈ Just one of C21,C21 because disjoint by definition

case 3 x∈C

You mean "x∈B3"

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

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

 

[Jairo Rojas]

You mean "x∈B1"You mean "x∈B2"You mean "x∈B3"

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