Question

how many elements must a set have if the number of proper subsets of the set 1/2 of the total number of subsets of the set?

Answer 1

If a set S has n-number of elements, then the TOTAL NUMBER OF subsets it will have is
\[\huge 2^n\]However, one of these is the set S itself. Therefore, the total number of PROPER SUBSETS a set with n elements would have is
\[\huge 2^n - 1\]

Answer 2

And is this supposedly equal to
\[\huge \frac12 \cdot 2^n\]

Answer 3

So...
\[\huge 2^n - 1 = \frac12 \cdot 2^n\]and solve for n.

Answer 4

doing this equation will solve for n?

Answer 5

Yes in effect. Pro-tip... let \(\large u = 2^n\) first, and solve for u.

Answer 6

1

Answer 7

?

Answer 8

Yes.

Answer 9

so i would do 2^1-1=1/2*2^1?

Answer 10

No, you've already solved for n... that's the number of elements your set has...

Answer 11

ohhhh

Answer 12

But yeah, you'll find that if you let n = 1, the equation checks out...

Answer 13

It is in fact, the only value for n, which would make the equation true.