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已知集合 $M \subseteq\left\{x \mid x=i^{n}+\mathrm{i}^{-n}, n \in \mathrm{N}_{+}\rig...

Mar 8, 2024

已知集合

M \subseteq\left\{x \mid x=i^{n}+\mathrm{i}^{-n}, n \in \mathrm{N}_{+}\right\}

(其中

\mathrm{i}

为虚数单位)，则满足条件的集合

M

的个数为

Solution by Steps

step 1

Identify the pattern of the powers of the imaginary unit i

step 2

Recall that

i^2 = -1

i^3 = -i

i^4 = 1

, and the powers of i repeat every four terms

step 3

Calculate the expression

i^n + i^{-n}

for the first few positive integers to find a pattern

step 4

Observe that

i^n + i^{-n}

will always be real because the imaginary parts cancel out

step 5

Note that for

n = 4k

and

n = 4k+2

(where k is a non-negative integer), the expression

i^n + i^{-n}

equals 2 or -2, respectively

step 6

For

n = 4k+1

and

n = 4k+3

, the expression

i^n + i^{-n}

equals 0

step 7

Determine the possible values for x in the set M

step 8

Since the possible values for x are 2, -2, and 0, the subsets of M can include any combination of these values

step 9

Count the number of subsets of M, including the empty set

step 10

There are 2^3 = 8 possible subsets of M, including the empty set

Answer

Key Concept

Powers of the imaginary unit and subset counting

Explanation

The powers of i repeat every four terms, and the sum

i^n + i^{-n}

will only yield the values 2, -2, or 0. The number of subsets of a set with three distinct elements is

2^3

, including the empty set.