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?

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.

in the math equation 8 x 7 = 56, the 8 and the 7 are both factors of 56, since in this lesson we will use square tiles to figure out the factors of a whole find as many different arrays as you can, but follow one important rule: all the rows must have the i can make two rows of nine, so 2 and 9 are both factors of 18.

step-by-step explanation: