Let X be a set whose elements are sets, and consider the subset relation ⊆ on X, i.e. for two elements A; B ∈ X, the pair (A; B) is an element of the relation if and only if A is a subset of B

Prove that ⊆ is a partial order on X.

so I have to prove it is reflective ,antisymmetric and transitive.....

a≤a reflective ...every element is related to itself

a≤b and b≤a then a = b so that is antisymmetric

a≤b and b≤c then a≤c transitive......

I'm not sure how to do this ...

Let X be any collection of sets and define the subset relation ⊆ on X as follows: For all A,B∈ X A⊆ B ⇔ for all X, if X ∈ A then x ∈ B.

Show that ⊆ is a partial order?

- Reflexive For ⊆ to be reflexive means that A ⊆ A . A ⊆ Ameans that A ⊂ A or A= A and A=A is always true.

Transitive

A ⊆ B & B ⊆ C⇒ U ⊆ C

A ⊆ B ⇔ for all X, if X ∈ A then X ∈ B.

B ⊆ C ⇔ for all Y, if Y ∈ B then Y ∈ C.

Let X be an arbitrary element of A ⇒ X ∈ B (definition of subset) ⇒ X ∈ C. Since this is true for an arbitrary element of A, it is true of all elements of A ⇒ A ⊆ C - Antisymmetric

For ⊆ to be antisymmetric means that for all sets A and B in A if A ⊆ B & B ⊆ A then A=C. Which is true by definition of equality of sets.

Is that ok my answer? please help...