What is the number of proper subsets for a well-defined set?

For example:-

A={1,2,3}

Some say that the answer is \(8\), because number of proper subsets is given by the formula \(2^{n(A)}\) because all the possible combinations are

{\(\phi\), {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}}.

Some say the answer is \(7\), because \(\phi\) is a void set and thus, it is not a proper subset. So, the possible combinations are

{{1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}}.

Some say the answer is \(6\), because \(\phi\) is a void set and {1,2,3} is the set itself, thus, they are not proper subsets. So, the possible combinations are

{{1}, {2}, {3}, {1,2}, {2,3}, {1,3}}.

What do you believe?

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## Comments

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TopNewestI believe that the answer is 7 but for a different reason. A proper subset is cannot include the set itself. These are the subsets {\(\phi\),{1},{2},{3},{1,2},{2,3},{1,3}}

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Good, but I believe the third one :O

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