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

And altogether we gain 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 methods we have the right to choose the items (the order of the items does not matter), consisting of choosing none, or all.

You are watching: Power set of 1 2 3 4


Example: The shop has actually banana, cacao and also lemon ice cream.

Whatexecute you order?

Nothing at all: Or possibly just banana: banana. Or just chocolate or just lemonOr 2 together: banana,chocolate or banana,lemon or cacao,lemonOr all three! banana, cocoa,lemon

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


Example: a,b,c has actually three members (a,b and also c).

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


Notation

The variety of members of a collection is often created as |S|, so once S has n members we have the right to write:


Example: for the collection S=1,2,3,4,5 exactly how many type of members will certainly the power collection have?

Well, S has actually 5 members, so:

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


You will certainly view in a minute why the variety of members is a power of 2

It"s Binary!

And right here is the many remarkable thing. To produce the Power Set, create down the sequence of binary numbers (making use of n digits), and also then let "1" intend "put the corresponding member into this subset".

So "101" is reput by 1 a, 0 b and 1 c to obtain us a,c

Like this:


Well, they are not in a pretty order, however they are all tright here.

Another Example

Let"s eat! We have 4 seasonings of ice cream: banana, cacao, lemon, and strawberry. How many different ways deserve to we have them?

Let"s usage letters for the flavors: b, c, l, s. Example selections include:

(nopoint, you are on a diet)b, c, l, s (eincredibly flavor)b, c (banana and also cacao are excellent together)etcLet"s make the table utilizing "binary":
bclsSubset
00000
10001s
20010l
30011l,s
...... etc ..... and so on ...
121100b,c
131101b,c,s
141110b,c,l
151111b,c,l,s

And the outcome 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


*

Symmetry

In the table above, did you notice that the initially subset is empty and the last has actually eextremely member?

But did you additionally notice that the second subset has "s", and the second last subset has every little thing except "s"?

*

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

This is bereason the binary numbers (that we offered to aid us obtain all those combinations) have actually a beautiful and also elegant pattern.


A Prime Example

The Power Set deserve to be valuable in unmeant areas.

I wanted to discover all components (not simply the prime determinants, but all factors) of a number.

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

That took a lengthy time for huge numbers.

But might I try to integrate the prime factors?

Let me watch, the prime components of 510 are 2×3×5×17 (making use of prime factor tool).

So, all the factors of 510 are:

2, 3, 5 and also 17,2×3, 2×5 and 2×17 also, and 2×3×5 and 2×3×17 and ... .. aha! Similar to ice cream I necessary a Power Set!

And this is what I got:


2,3,5,17SubsetFactors of 510
00000 1
100011717
2001055
300115,175 × 17 = 85
4010033
501013,173 × 17 = 51
... and so on ...... and so on ...... and so on ...
1511112,3,5,172 × 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 also −1, −2, −3, etc as well). See the All Factors Device.

Automated

I couldn"t resist making Power Sets obtainable to you in an automated means.

See more: What Is A Bodegon What Is Its Significance ? What Is A Bodegon

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


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