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Thread: Partial order question proof.....

  1. #1
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    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.

    Is that ok my answer? please help...
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  2. #2
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    Re: Partial order question proof.....

    it's a mess but you seem to have stated the main points.
    Thanks from bee77
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  3. #3
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    Re: Partial order question proof.....

    Quote Originally Posted by bee77 View Post
    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 $A\subset C$?
    Thanks from bee77
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    Re: Partial order question proof.....

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