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# Number of proper subsets

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?

Note by Ashish Siva
1 year, 5 months ago

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I 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}} · 1 year, 5 months ago