Or any two flavors: banana, chocolate, banana, vanilla, or chocolate, vanilla,
Or every three spices (no the isn"t greedy),
Or you could say "none at every thanks", i m sorry is the "empty set":
Example: The set alex, billy, casey, dale
Has the subsets:alexbillyetc ...
You are watching: How many subsets does a set of n elements have
It likewise has the subsets:alex, billyalex, caseybilly, daleetc ...
Also:alex, billy, caseyalex, billy, daleetc ...
And also:the whole set: alex, billy, casey, dalethe empty set:
Now let"s begin with the Empty collection and move on increase ...
How countless subsets does the empty collection have?
You can choose:the whole set: the empty set:
But, hang on a minute, in this case those are the exact same thing!
So theempty set really has just 1 subset (whichis itself, the empty set).
It is choose asking "There is nothing available, for this reason what execute you choose?" prize "nothing". The is your just choice. Done.
ASet with One Element
The set could it is in anything, but let"s just say the is:
How countless subsets walk the collection apple have?the totality set: applethe north set:
And that"s all.Youcanchoose the one element, or nothing.
So any collection with one element will have actually 2 subsets.
ASet v Two Elements
Let"s add another element to our example set:
How numerous subsets walk the collection apple, banana have?
It can have apple, or banana, and don"t forget:the entirety set: apple, bananathe empty set:
So a collection with two elements has 4 subsets.
ASet With three Elements
apple, banana, cherry
OK, let"s be more systematic now, and list the subsets by how many facets they have:
Subsets v one element: apple, banana, cherry
Subsets through two elements: apple, banana, apple, cherry, banana, cherry
And:the whole set: apple, banana, cherrythe north set:
In truth we can put the in a table:
|List||Number of subsets|
|one element||apple, banana, cherry||3|
|two elements||apple, banana, apple, cherry, banana, cherry||3|
|three elements||apple, banana, cherry||1|
(Note: did you view a sample in the numbers there?)
Setswith Four aspects (Your Turn!)
Now try to perform the same for this set:
apple, banana, cherry, date
Here is a table for you:
|List||Number the subsets|
(Note: if friend did this right, there will certainly be a sample to the numbers.)
Setswith five Elements
apple, banana, cherry, date, egg
Here is a table for you:
|List||Number the subsets|
(Was over there a sample to the numbers?)
Setswith 6 Elements
apple, banana, cherry, date, egg, fudge
OK ... We don"t need to finish a table, because...
How countless subsets space there because that a collection of 6 elements? _____How countless subsets space there because that a collection of 7 elements? _____
Now let"s think around subsets and sizes:Theemptyset hasjust 1subset: 1A set with one facet has 1 subset with no elements and 1subset v one element: 1 1A collection with twoelements has actually 1 subset v no elements, 2 subsets v one element and also 1 subset with two elements: 12 1A collection with threeelements has actually 1 subset with no elements, 3 subsets with oneelement, 3 subsets with two elements and also 1 subset through threeelements: 1 3 3 1and therefore on!
Do you identify thispattern that numbers?
They are the number from Pascal"sTriangle!
This is very useful, since now friend can inspect if you have the right variety of subsets.
Note: the rows start at 0, and an in similar way the columns.
Example: because that the set apple, banana, cherry, date, egg you perform subsets of length three:apple, banana, cherryapple, banana, dateapple, banana, eggapple, cherry, egg
But the is only 4 subsets, how numerous should over there be?
Well, friend are selecting 3 the end of 5, so go to row 5, place 3 the Pascal"s Triangle (remember to start counting at 0) to uncover you require 10 subsets, for this reason you must think harder!
In fact these are the results: apple,banana,cherry apple,banana,date apple,banana,egg apple,cherry,date apple,cherry,egg apple,date,egg banana,cherry,date banana,cherry,egg banana,date,egg cherry,date,egg
Calculating The Numbers
Is over there a means of calculating the numbers such as 1, 4, 6, 4 and also 1 (instead of feather them up in Pascal"s Triangle)?
Yes, we can discover the variety of ways of choosing each number ofelements utilizing Combinations.
There space four aspects in the set, and:
The number of ways ofselecting 0 aspects from 4 = 4C0 = 1The number of ways ofselecting 1 element from 4 = 4C1 = 4The variety of ways of picking 2 aspects from 4 = 4C2 = 6The variety of ways of choosing 3 facets from 4 = 4C3 = 4The number of ways of choosing 4 aspects from 4 = 4C4 = 1 full number ofsubsets = 16
The variety of waysofselecting 0 aspects from 5 = 5C0 = 1The variety of ways ofselecting 1 facet from 5 = ___________The number of ways of selecting 2 facets from 5 = ___________The variety of ways of choosing 3 facets from 5 = ___________The variety of ways of choosing 4 aspects from 5 = ___________Thenumber of ways of picking 5 aspects from 5 = ___________ Total variety of subsets = ___________
In this activity you have:Discovered a ascendancy fordetermining the total variety of subsets for a provided set: A set with nelements has actually 2n subsets.Found a connection betweenthe number of subsets that each dimension with the numbers in Pascal"striangle.Discovered a quick method tocalculate this numbers using Combinations.
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Moreimportantly you have learned how various branches of mathematics canbe merged together.