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Well Founded Sets

  1. Apr 5, 2010 #1
    I am struggling to properly understand the concept of a well-founded set.

    Is this well founded, d = {{x},{x,y},{x,y,z}}

    because there exists an element of d i.e. {x} = e

    such that d n e = 0 ???
  2. jcsd
  3. Apr 5, 2010 #2
    Well-foundness usually is related to a relation (order).

    Well-foundness is generalization of well-order. The difference is that well-order is linear and well-found is not necessary linear, but is partial order.

    The definition is:
    The relation E on set P is well-founded if any non-empty subset has E-minimal element.
    Now, very often the natural order(relation) for sets is belonging [tex]\in[/tex].

    So in this case, in your example d is well-founded since any non-empty subset:
    1){{x}},2) {{x,y}},3) {{x,y,z}}, 4){{x},{x,y}}, 5){{x},{x,y,z}}, 6){{x,y}, {x,y,z}} , 7){{x},{x,y},{x,y,z}} has a minimal elements such as:
    1){x}, 2){x,y}, 3){x,y,z}, 4){x}, {x,y} (since {x} [tex]\notin[/tex] {x,y} 5) {x}, {x,y,z} 6) {x,y} , {x,y,z} 7) {x}, {x,y} {x,y,z}

    Notice that [tex]\in[/tex] is partial order on your set d. That is why 4), 5) 6) and 7) have several minimal elements.

    So d is well-founded but not well-ordered, since it is not linear ordered by [tex]\in[/tex].
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