What is an example of: A is a subset of B and B is a subset C, but C is not a subset of A?

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Actually, such a situation is typical. If `A` is a subset of `B` and `B` is a subset of `C,` then `A` is a subset of `C` (any element of `A` is an element of `B` and thus is an element of `C`).

Therefore for `C` to be a subset of `A,` `A` and `C` must coincide. And for `C` not to be a subset of `A` it is sufficient that `B` has at least one extra element compared to `A,` or `C`  has at least one extra element compared to `B.` This is easy to achieve.

For example, let `A = {1},` `B = {1,2}` and `C={1,2,3}.` Then all the conditions are satisfied: `A sub B sub C,`  but not `C sub A.`

Or `A = NN,` `B = ZZ,` `C = RR.`

 

 

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