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candymorgan81
06.09.2019 •
Mathematics
Let n be a positive integer and define [n] to be the set of the first n positive integers. that is, [n] = {1, 2, 3, . . , n}. we want to select two disjoint, possibly empty subsets a, b of [n]. in how many ways can we do this?
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Ответ:
There are
ways of doing this
Hi!
To solve this problem we can think in term of binary numbers. Let's start with an example:
n=5, A = {1, 2 ,3}, B = {4,5}
We can think of A as 11100, number 1 meaning "this element is in A" and number 0 meaning "this element is not in A"
And we can think of B as 00011.
Thinking like this, the empty set is 00000, and [n] =11111 (this is the case A=empty set, B=[n])
This representation is a 5 digit binary number. There are
of these numbers. Each one of this is a possible selection of A and B. But there are repetitions: 11100 is the same selection as 00011. So we have to divide by two. The total number of ways of selecting A and B is the
.
This can be easily generalized to n bits.
Ответ: