What do you call the process of choosing a subset of a set?

A Power Set is a set of all the subsets of a set.

Table of Contents Show

All The Subsets
Example: The shop has banana, chocolate and lemon ice cream.
How Many Subsets
It's Binary!
Another Example
A Prime Example

OK? Got that? Maybe an example will help...

All The Subsets

For the set {a,b,c}:

The empty set {} is a subset of {a,b,c}
And these are subsets: {a}, {b} and {c}
And these are also subsets: {a,b}, {a,c} and {b,c}
And {a,b,c} is a subset of {a,b,c}

And altogether we get the Power Set of {a,b,c}:

P(S) = { {}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} }

Think of it as all the different ways we can select the items (the order of the items doesn't matter), including selecting none, or all.

Example: The shop has banana, chocolate and lemon ice cream.

What do you order?

Nothing at all: {}
Or maybe just banana: {banana}. Or just {chocolate} or just {lemon}
Or two together: {banana,chocolate} or {banana,lemon} or {chocolate,lemon}
Or all three! {banana, chocolate,lemon}

Question: if the shop also has strawberry flavor what are your options? Solution later.

How Many Subsets

Easy! If the original set has n members, then the Power Set will have 2n members

So, the Power Set should have 23 = 8, which it does, as we worked out before.

Notation

The number of members of a set is often written as |S|, so when S has n members we can write:

|P(S)| = 2n

Well, S has 5 members, so:

|P(S)| = 2n = 25 = 32

You will see in a minute why the number of members is a power of 2

It's Binary!

And here is the most amazing thing. To create the Power Set, write down the sequence of binary numbers (using n digits), and then let "1" mean "put the matching member into this subset".

So "101" is replaced by 1 a, 0 b and 1 c to get us {a,c}

Like this:

	abc	Subset
0	000	{ }
1	001	{c}
2	010	{b}
3	011	{b,c}
4	100	{a}
5	101	{a,c}
6	110	{a,b}
7	111	{a,b,c}

Well, they are not in a pretty order, but they are all there.

Another Example

Let's eat! We have four flavors of ice cream: banana, chocolate, lemon, and strawberry. How many different ways can we have them?

Let's use letters for the flavors: {b, c, l, s}. Example selections include:

{} (nothing, you are on a diet)
{b, c, l, s} (every flavor)
{b, c} (banana and chocolate are good together)
etc

Let's make the table using "binary":

	bcls	Subset
0	0000	{}
1	0001	{s}
2	0010	{l}
3	0011	{l,s}
...	... etc ..	... etc ...
12	1100	{b,c}
13	1101	{b,c,s}
14	1110	{b,c,l}
15	1111	{b,c,l,s}

And the result is (more neatly arranged):

P = { {}, {b}, {c}, {l}, {s}, {b,c}, {b,l}, {b,s}, {c,l}, {c,s}, {l,s}, {b,c,l}, {b,c,s},
{b,l,s}, {c,l,s}, {b,c,l,s} }

In the table above, did you notice that the first subset is empty and the last has every member?

But did you also notice that the second subset has "s", and the second last subset has everything except "s"?

In fact when we mirror that table about the middle we see there is a kind of symmetry.

This is because the binary numbers (that we used to help us get all those combinations) have a beautiful and elegant pattern.

A Prime Example

The Power Set can be useful in unexpected areas.

I wanted to find all factors (not just the prime factors, but all factors) of a number.

I could test all possible numbers: I could check 2, 3, 4, 5, 6, 7, etc...

That took a long time for large numbers.

But could I try to combine the prime factors?

Let me see, the prime factors of 510 are 2×3×5×17 (using prime factor tool).

So, all the factors of 510 are:

2, 3, 5 and 17,
2×3, 2×5 and 2×17 as well, and
2×3×5 and 2×3×17 and ...
.. aha! Just like ice cream I needed a Power Set!

And this is what I got:

	2,3,5,17	Subset	Factors of 510
0	0000	{ }	1
1	0001	{17}	17
2	0010	{5}	5
3	0011	{5,17}	5 × 17 = 85
4	0100	{3}	3
5	0101	{3,17}	3 × 17 = 51
	... etc ...	... etc ...	... etc ...
15	1111	{2,3,5,17}	2 × 3 × 5 × 17 = 510

And the result? The factors of 510 are 1, 2, 3, 5, 6, 10, 15, 17, 30, 34, 51, 85, 102, 170, 255 and 510 (and −1, −2, −3, etc as well). See the All Factors Tool.

Automated

I couldn't resist making Power Sets available to you in an automated way.

So, when you need a power set, try Power Set Maker.

One solution I was considering was to choose items for I' successively such that the average p value of the items selected so far always falls within some upper and lower bound.

Obviously, we could just choose p_l and p_u as our lower and upper bounds, respectively, and this would satisfy the average p value constraint. But this would also be too restrictive. It could leave out a large number of possible I' where small p values might balance out high p values.

So I was considering letting the upper and lower bounds start out far apart when the first items are selected and then let them narrow down to p_l and p_u when the final mth item is selected. So we would have a cone of ranges that would become narrower as more items are picked.

More precisely, we would define two functions u and l such that the average of the first i items picked (call this p_i) would satisfy l(i) <= p_i <= u(i) for 0 <= i <= m.

I'm not really sure how to define these two functions, though. I know that l(m) = p_l and u(m) = p_u, but I don't know what l(0) or u(0) should be. Also, should the functions be linear?

Comments or suggestions about this possible solution are welcome.

Example 1: Given A = {1, 2, 4} and B = {1, 2, 3, 4, 5}, what is the relationship between these sets?

We say that A is a subset of B, since every element of A is also in B. This is denoted by:

A Venn diagram for the relationship between these sets is shown to the right.

Answer: A is a subset of B.

Another way to define a subset is: A is a subset of B if every element of A is contained in B. Both definitions are demonstrated in the Venn diagram above.

Example 2: Given X = {a, r, e} and Y = {r, e, a, d}, what is the relationship between these sets?

We say that X is a subset of Y, since every element of X is also in Y. This is denoted by:

A Venn diagram for the relationship between these sets is shown to the right.

Answer: X is a subset of Y.

Example 3: Given P = {1, 3, 4} and Q = {2, 3, 4, 5, 6}, what is the relationship between these sets?

We say that P is not a subset of Q since not every element of P is not contained in Q. For example, we can see that 1 Q. The statement "P is not a subset of Q" is denoted by:

Note that these sets do have some elements in common. The intersection of these sets is shown in the Venn diagram below.

Answer: P is not a subset of Q.

The notation for subsets is shown below.

Symbol	Meaning
	is a subset of
	is not a subset of

Example 4: Given A = {1, 2, 3, 4, 5} and B = {3, 1, 2, 5, 4}, what is the relationship between A and B?

Analysis: Recall that the order in which the elements appear in a set is not important. Looking at the elements of these sets, it is clear that:

Answer: A and B are equivalent.

Definition: For any two sets, if A

B and B

A, then A = B. Thus A and B are equivalent.

Example 5: List all subsets of the set C = {1, 2, 3}.

Answer:

Subset	List all possible combinations of elements...
D = {1}	one at a time
E = {2}	one at a time
F = {3}	one at a time
G = {1, 2}	two at a time
M = {1, 3}	two at a time
N = {2, 3}	two at a time
P = {1, 2, 3}	three at a time
Ø	The null set has no elements.

Looking at example 5, you may be wondering why the null set is listed as a subset of C. There are no elements in a null set, so there can be no elements in the null set that aren't contained in the complete set. Therefore, the null set is a subset of every set. You may also be wondering: Is a set a subset of itself? The answer is yes: Any set contains itself as a subset. This is denoted by:

A subset that is smaller than the complete set is referred to as a proper subset. So the set {1, 2} is a proper subset of the set {1, 2, 3} because the element 3 is not in the first set. In example 5, you can see that G is a proper subset of C, In fact, every subset listed in example 5 is a proper subset of C, except P. This is because P and C are equivalent sets (P = C). Some mathematicians use the symbol

to denote a subset and the symbol

to denote a proper subset, with the definition for proper subsets as follows:

If A

B, and A ≠ B, then A is said to be a proper subset of B and it is denoted by A

While it is important to point out the information above, it can get a bit confusing, So let's think of subsets and proper subsets this way:

Subsets and Proper Subsets

The set {1, 2} is a proper subset of the set {1, 2, 3}.

The set {1, 2, 3} is a not a proper subset of the set {1, 2, 3}.

Do you see a pattern in the examples below?

Example 6: List all subsets of the set R = {x, y, z}. How many are there?

Subsets

D = {x}

E = {y}

F = {z}

G = {x, y}

H = {x, z}

J = {y, z}

K = {x, y, z}

Answer: There are eight subsets of the set R = {x, y, z}.

Example 7: List all subsets of the set C = {1, 2, 3, 4}. How many are there?

Subsets
D = {1}	M = {2, 4}
E = {2}	N = {3, 4}
F = {3}	O = {1, 2, 3}
G = {4}	P = {1, 2, 4}
H = {1, 2}	Q = {1, 3, 4}
J = {1, 3}	R = {2, 3, 4}
K = {1, 4}	S = {1, 2, 3, 4}
L = {2, 3}	Ø

Answer: There are 16 subsets of the set C = {1, 2, 3, 4}.

In example 6, set R has three (3) elements and eight (8) subsets. In example 7, set C has four (4) elements and 16 subsets. To find the number of subsets of a set with n elements, raise 2 to the nth power: That is:

The number of subsets in set A is 2n , where n is the number of elements in set A.

L e s s o n S u m m a r y

Subset: A is a subset of B: if every element of A is contained in B. This is denoted by A

Equivalent Sets: For any two sets, if A

B and B

A, then A = B.

Null set: The null set is a subset of every set.

Sets and subsets: Any set contains itself as a subset. This is denoted by A

Proper Subsets: If A

B, and A ≠ B, then A is said to be a proper subset of B and it is denoted by A

Number of Subsets: The number of subsets in set A is 2n , where n is the number of elements in set A.

Exercises

Directions: Read each question below. Select your answer by clicking on its button. Feedback to your answer is provided in the RESULTS BOX. If you make a mistake, rethink your answer, then choose a different button.