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Answers
Answer:
Easy! If the original set has n members, then the Power Set will have 2n members
Example: {a,b,c} has three members (a,b and c).
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
Example: for the set S={1,2,3,4,5} how many members will the power set have?
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
ice cream
Let's eat! We have four flavours of ice cream: banana, chocolate, lemon, and strawberry. How many different ways can we have them?
Let's use letters for the flavours: {b, c, l, s}. Example selections include:
{} (nothing, you are on a diet)
{b, c, l, s} (every flavour)
{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} }
yin yang
Symmetry
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"?
binary symmetry
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.
Step-by-step explanation:
hope it helps