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?

#Combinatorics

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Comments

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}}

A Former Brilliant Member - 5 years, 1 month ago

Good, but I believe the third one :O

Ashish Menon - 5 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$