If a set has 1024 subsets, how many elements does it contain?

To determine how many elements a set has based on the number of its subsets, we can use a fundamental concept from set theory. The number of subsets of a set is calculated using the formula:

Number of subsets = 2ⁿ

where n is the number of elements in the set.

If we are given that the set has 1024 subsets, we can set up the equation:

2ⁿ = 1024

Next, we need to express 1024 as a power of 2. By calculation, we find:

1024 = 2¹⁰

Now we can equate the powers:

2ⁿ = 2¹⁰

This implies that:

n = 10

Thus, the set contains 10 elements.

In summary, if a set has 1024 subsets, it must contain 10 elements.