# Thread: Partial order question proof.....

1. ## Partial order question proof.....

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.

2. ## Re: Partial order question proof.....

it's a mess but you seem to have stated the main points.

3. ## Re: Partial order question proof.....

Originally Posted by bee77
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...
@Bee77, part of your problem is vocabulary. We say that $\mathscr{R}$ is a partial ordering of $\mathcal{P}(X)$, the power set of $X$ iff $\mathscr{R}$ is a relation on $X$ which is reflexive, antisymmetric, & transitive.
1. Is each subset of $X$ a subset of itself?
2. If each of $A~\&~B$ is a subset of $X$ and $A\subset B~\&~B\subset A$ is it true that $A=B$?
3. If each of $A~,~B,~\&~C$ is a subset of $X$ and $A\subset B~\&~B\subset C$ is it true that $A\subset C$?

4. ## Re: Partial order question proof.....

Proofs are definitely a weak point of mine ....thanks for the input