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A set is not allowed to has itself as a member: [tex]X \notin X[/tex]. But I wonder if this is allowed: [tex]\{X\} \in X[/tex].
Outlined said:A set is not allowed to has itself as a member: [tex]X \notin X[/tex]. But I wonder if this is allowed: [tex]\{X\} \in X[/tex].
CRGreathouse said:Neither are allowed in ZF. But there are theories in which both are allowed -- for example, ZF without the Axiom of Foundation.
A set containing itself is a mathematical concept where a set contains at least one element that is also a set. In other words, the set is an element of itself.
A set containing itself is important because it allows for the creation of more complex mathematical structures and helps to define new concepts such as recursion and self-reference.
A regular set does not contain itself as an element, while a set containing itself does. This creates a unique relationship between the set and its elements.
No, a set can only contain itself once. If it were to contain itself multiple times, it would essentially become an infinite loop and not be a well-defined mathematical concept.
One example is a set of subsets, where each subset is a set of elements from the original set. Another example is a family tree, where the set of parents contains itself as an element when a parent is also a grandparent to their own children.