The elements of X are sets.

Consider the subset relation R ("⊆") on X. (A, B) ∈ R iff A is a subset (⊆) of B.

1. Give an example with |X| = 4 where relation "⊆" is a total order.

This relation R is a Partial Order (POSET).

My attempt...

X = { {}, {1}, {1, 2}, {1, 2, 3} }. I'm not sure if this is correct.

2. Give an example with |X| = 4 where the subset relation R does not contain any pair (A, B) with A ≠ B.

I'm currently lost on this.

3. Give an example with |X| = 6 where the subset relation does not contain any pair (A, B) with A ≠ B and all elements of X are subsets of the set {1, 2, 3, 4}.

Lost on this, too.

Cheers for assistance.

Originally Posted by Induction
The elements of X are sets.

Consider the subset relation R ("⊆") on X. (A, B) ∈ R iff A is a subset (⊆) of B.

1. Give an example with |X| = 4 where relation "⊆" is a total order.

This relation R is a Partial Order (POSET).

My attempt...

X = { {}, {1}, {1, 2}, {1, 2, 3} }. I'm not sure if this is correct.

Yes, that is correct.

2. Give an example with |X| = 4 where the subset relation R does not contain any pair (A, B) with A ≠ B.

I'm currently lost on this.
It pretty much has to be "singleton" sets such as {{1}, {2}, {3}, {4}}.

3. Give an example with |X| = 6 where the subset relation does not contain any pair (A, B) with A ≠ B and all elements of X are subsets of the set {1, 2, 3, 4}.

Lost on this, too.

Try {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}. No set is a subset of any other.

Cheers for assistance.