Question

Prove that a set with n elements has 2n subsets.

Answer 1

The number of electrons in the shell equals 2n² and in each subshell is 2(2l + 1), derived from the quantum mechanical principles and the Pauli exclusion principle governing electron arrangement in atoms.

Step-by-step explanation:

To prove that the number of electrons in the shell equals 2n² and that the number in each subshell is 2(2l + 1), we need to use the quantum mechanical principles that govern the arrangement of electrons in an atom. According to the quantum model, each electron in an atom is described by four quantum numbers: n, l, mi, and ms. The principal quantum number n represents the shell level, the angular momentum quantum number l describes the subshell (with values ranging from 0 to n-1), the magnetic quantum number mi describes the orientation of the subshell (ranging from -l to +l), and the spin quantum number ms indicates the electron's spin (which can be +1/2 or -1/2).

Each shell level n can have subshell values from 0 to n-1. For each value of l, there are 2l+1 possible values for mi, and for each mi, there can be two electrons (one with spin up and one with spin down, according to the Pauli exclusion principle). Therefore, the maximum number of electrons in any subshell is 2(2l+1). Summing over all values of l will give the total number of electrons in a shell, which is 2n². This follows from considering all possible orientations and spin states for each value of l within a shell. For the n=2 shell as an example, there are l=0 and l=1 subshells. The s subshell (l=0) can hold 2 electrons, and the p subshell (l=1) can hold 6 electrons, for a total of 2(2^2) = 8 electrons in the n=2 shell. Using similar calculations for other values of n will confirm the general formula.

The application of the Pauli exclusion principle ensures that no two electrons can have the same set of all four quantum numbers, which fundamentally limits the number of possible electrons in a subshell and a shell.

Answer 2

From a set of
`n` elements, there is only one subset that can be chosen that contains no elements (the empty set), i.e.
`\dbinom n0`.

(
`\dbinom nk=(n!)/(k!(n-k)!)` denotes the binomial coefficient)

Similarly, there are
`\dbinom n1` possible subsets consisting of only one element that can be taken from the larger set,
`\dbinom n2` subsets consisting of two members, and so on, which means the total number of possible subsets that can be chosen is


`\displaystyle\binom n0+\binom n1+\cdots+\binom n{n-1}+\binom nn`

`=\displaystyle\sum_(k=0)^n\binom nk`

Writing this as


`=\displaystyle\sum_(k=0)^n\binom nk1^(n-k)1^k`

we can apply the binomial theorem, which states that this is equivalent to
`(1+1)^n=2^n`.