# Binary Sequences

Let $B_n$ = $\mathcal{P}(\{1, 2, \dots, n\})$.

The set $\{0,1\}^n = \{a_1, a_2, ... , a_n : a_i \in \{0,1\}\}$ is called the length of binary sequences of length $n$.

I want to verify and work on the following questions:

a) Describe a function $f:\{0,1\}^{n} \to B_n$ (which is a bijection), and give its inverse.

My attempt was as follows: Because this is a bijection, this would imply a one-to-one correspondence between the domain and the target set. Therefore, this function maps maps a binary sequence of length $n$ to the power set of length $n$.

I assumed that $f^{-1}: B_n \to \{0,1\}^{n}$ would be true.

b) Using part a, determine $|B_n|$.

Because $B_n$ = power set of $\{1, 2, \dots , n\}$, I claimed that the cardinality would be $2^n$.

c) Let $S_k$ be the set of elements of $\{0,1\}^{n}$ which have exactly $k$ coordinates equal to 1. Determine the range of the restriction of $f$ to $S_k$.

d) Determine $|S_k|$.

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For part a, you're asked to find a bijection, not to describe what a bijection is. What do you think would be a good correspondence between the power set and the set of binary strings? – EuYu Nov 5 '12 at 4:38

a) I think here "describe" means "give a specific example of". Hint: for a given subset $S$ of $\{1,\dots,n\}$, an element of $\{1,\dots,n\}$ is either in $S$ or not in $S$.

b) That's correct.

c) For this you need part a). What is the size of the set $f([a_1,\dots,a_n])$ if there are $k$ coordinates switched "on"?

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Dear wj32, I'm having difficulty understanding the idea of 'k coordinates equal to 1' and being 'switched on'. Could you elaborate this? – Julian Park Nov 5 '12 at 4:53
@JulianPark: I'm assuming you've found the bijection in part a). Let $S=f([a_1,\dots,a_n])$. Each bit $a_i$ decides whether the number $i$ is to be included in $S$. If exactly $k$ of the $a_i$ are set to 1, then the size of $S$ is $k$. Therefore the answer to part c) is "all subsets of $\{1,\dots,n\}$ of size $k$". – wj32 Nov 5 '12 at 5:10

Hint: You are in a cafeteria, at the end of the lunch hour. There is one Type $1$ item left, and one Type $2$ item, and one Type $3$ item, and so on up to one Type $n$ item. You are allowed to put on your tray $0$ or $1$ Type $1$ items, $0$ or $1$ type $2$ items, and so on.

You decide to plan lunch ahead of time, by writing down a sequence of $0$'s and/or $1$'s. So for example if $n=5$, and you write the sequence $01101$, that means you plan to say no to Item $1$, yes to Items $2$ and $3$, no to $4$, and yes to $5$. Note that the sequence $00000$ is a valid one: maybe you are on a diet.

Every sequence of $n$ $0$'s and or $1$'s represents a choice of meal. (That includes the empty meal.)

For (c), a sequence is in $S_k$ if and only if you have decided to have a $k$-item meal. For (d), how many ways are there to choose $k$ items for your meal, from the $n$ items available?

If you are not hungry, you are a teacher. The $n$ students are lined up in a row. You decide to choose "some" of them (possibly all, possibly none) to get a prize. You write down a sequence of $0$'s and/or $1$'s to record which students will get a prize. So for example if the line has $n=5$ students, $01101$ means the first student won't get a prize, the next two will, and so on. Each sequence uniquely determines the subset of the students who will get a prize.

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Choosing k items for your meal from n items available would be n choose k (sampling without replacement)? – Julian Park Nov 5 '12 at 5:02
@JulianPark: Yes, there are exactly $\binom{n}{k}$ ways to get a $k$-tem meal, just like there are $\binom{n}{k}$ ways to choose the $k$ positions that get a $1$. There are also, not coincidentally, $\binom{n}{k}$ subsets of size $k$. – André Nicolas Nov 5 '12 at 5:05

For a), let's say that $n=2$, so you have the following binary strings: $\{0,0\}$, $\{1,0\}$, $\{0,1\}$ and $\{1,1\}$. We know that the power set, $P(\{1,2\})$ has elements empty set, $\{1\}$, $\{2\}$ and $\{1,2\}$. They are bijective (both have $4$ elements).

Now, to describe the function.

$f:\{0,0\} \to \emptyset$,

$f:\{1,0\} \to \{1\}$,

$f:\{0,1\} \to \{2\}$,

$f:\{1,1\} \to \{1,2\}$.

So, we can see that the places with $1$ correspond to the element from the set of the power set (I'll call it set $A$) is being applied to. (i.e, $f$ maps $\{0,1\}$ to the second element of $A$, namely $\{2\}$).

In general, for $n$,

$f:\{0\}^n \to \emptyset$,

$f: \{1\} \cup \{0\}^{n-1} \to \{1\}$,

$f: \{0,1\} \cup \{0\}^{n-2} \to \{2\}$,

$\ldots$

I believe this is the best way to describe the bijective function.

The inverse when $n=2$ is simply:

$f^{-1}:\emptyset \to \{0,0\}$,

$f^{-1}:\{1\} \to \{1,0\}$,

$f^{-1}:\{2\} \to \{0,1\}$,

$f^{-1}:\{1,2\} \to \{1,1\}$.

For c), restricting $f$ by $S_k$, we know that if there are $k$ $1$'s, then there are $k$ elements from set $A$ in the $f$ by $S_k$. This means that there will only be sets with $k$ elements in them present in the restriction (i.e., if $k=2$ and $n=2$, then $\{1,1\}$ will be the only element of the restriction $f$ by $S_k$). So the range is all sets with $k$ elements, i.e., n choose k elements.

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