What are some examples where the intersection of two sets is a member of one of the sets?(adsbygoogle = window.adsbygoogle || []).push({});

Let [itex] A,B,C,D [/itex] be sets whose elements are sets of integers.

[tex] A = \{\emptyset, \{1,2\},\{3\} \} [/tex]

[tex] B = \{\{4,5,6\},\{7,8\} \} [/tex]

[tex] C = \{ \emptyset, \{7,8\} \} [/tex]

[tex] D = \{ \{3\}, \{4,5,6\} \} [/tex]

Then [itex] A \cap B = \emptyset [/itex] and taking the empty set to be unique (as mentioned in the recent thread https://www.physicsforums.com/threa...ms-specify-that-the-empty-set-is-open.773047/ ) this is the same empty set that is an element of [itex] A [/itex] so have that [itex] A \cap B \in A [/itex]

On the other hand [itex] A \cap C = \{\emptyset \} [/itex], which is to say that [itex] A \cap C [/itex] is a set with one element and that element is the empty set. This set is not the empty set, so we can't say that [itex] A \cap C \in A [/itex].

[itex] A \cap D = \{\{3\}\} [/itex]. My interpretation is that [itex] A \cap D [/itex] is a set with one member and that member is itself a set with one element, the integer 3. So [itex] \{\{3\}\} [/itex] is not a "set of integers", it is a "set of sets of integers". So we can't say that that [itex] A \cap D \in A [/itex].

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# Examples of A intersection B being an element of A

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