the cardinality of the power set of a = {r,s,t,u,w} is equal to what value?
Frequently times we are interested in the number of items in a prepare or subset. This is called the cardinality of the set.
Cardinality
The number of elements in a set is the cardinality of that prepare.
The cardinality of the set A is oft notated every bit |A| or n(A)
Instance 12
Let A = {1, 2, 3, iv, 5, 6} and B = {2, 4, half dozen, eight}.
What is the cardinality of B? A ⋃ B, A ⋂ B?
The cardinality of B is 4, since in that location are iv elements in the fix.
The cardinality of A ⋃ B is seven, since A ⋃ B = {1, two, 3, 4, v, half-dozen, 8}, which contains vii elements.
The cardinality of A ⋂ B is 3, since A ⋂ B = {2, 4, 6}, which contains 3 elements.
Example 13
What is the cardinality of P = the set of English language names for the months of the year?
The cardinality of this set is 12, since there are 12 months in the year.
Sometimes we may be interested in the cardinality of the union or intersection of sets, but not know the actual elements of each prepare. This is mutual in surveying.
Example 14
A survey asks 200 people "What beverage do you drinkable in the morning", and offers choices:
- Tea only
- Coffee merely
- Both coffee and tea
Suppose 20 report tea but, 80 report coffee only, xl study both. How many people drink tea in the morning? How many people beverage neither tea or coffee?
This question can most easily be answered past creating a Venn diagram. We can encounter that we can detect the people who drink tea by calculation those who drink only tea to those who drink both: 60 people.
We tin besides run into that those who beverage neither are those not contained in the whatever of the three other groupings, and so nosotros can count those by subtracting from the cardinality of the universal prepare, 200.
200 – 20 – 80 – 40 = 60 people who drink neither.
Case fifteen
A survey asks: Which online services have you lot used in the last calendar month:
- Take used both
The results prove 40% of those surveyed take used Twitter, 70% take used Facebook, and xx% have used both. How many people accept used neither Twitter or Facebook?
Let T be the ready of all people who have used Twitter, and F be the fix of all people who accept used Facebook. Notice that while the cardinality of F is 70% and the cardinality of T is forty%, the cardinality of F ⋃ T is not simply 70% + 40%, since that would count those who use both services twice. To notice the cardinality of F ⋃ T, we can add the cardinality of F and the cardinality of T, then subtract those in intersection that we've counted twice. In symbols,
n(F ⋃ T) = n(F) + n(T) – n(F ⋂ T)
due north(F ⋃ T) = lxx% + forty% – 20% = 90%
Now, to observe how many people take not used either service, nosotros're looking for the cardinality of (F ⋃ T)c . Since the universal set contains 100% of people and the cardinality of F ⋃ T = xc%, the cardinality of (F ⋃ T)c must exist the other 10%.
The previous case illustrated ii important properties
Cardinality properties
due north(A ⋃ B) = n(A) + n(B) – n(A ⋂ B)
north(Ac) = north(U) – due north(A)
Discover that the kickoff property tin can also be written in an equivalent grade past solving for the cardinality of the intersection:
due north(A ⋂ B) = n(A) + n(B) – n(A ⋃ B)
Example 16
Fifty students were surveyed, and asked if they were taking a social science (SS), humanities (HM) or a natural scientific discipline (NS) course the next quarter.
21 were taking a SS grade 26 were taking a HM course
19 were taking a NS course 9 were taking SS and HM
seven were taking SS and NS 10 were taking HM and NS
3 were taking all three seven were taking none
How many students are but taking a SS course?
It might aid to wait at a Venn diagram.
From the given information, we know that there are
three students in region e and
7 students in region h.
Since seven students were taking a SS and NS grade, nosotros know that n(d) + n(e) = vii. Since we know there are 3 students in region three, there must be
7 – 3 = iv students in region d.
Similarly, since in that location are 10 students taking HM and NS, which includes regions e and f, there must be
10 – three = vii students in region f.
Since ix students were taking SS and HM, in that location must be 9 – 3 = six students in region b.
Now, we know that 21 students were taking a SS grade. This includes students from regions a, b, d, and e. Since we know the number of students in all but region a, we can determine that 21 – six – 4 – 3 = 8 students are in region a.
eight students are taking only a SS course.
Try it Now iv
One hundred 50 people were surveyed and asked if they believed in UFOs, ghosts, and Bigfoot.
43 believed in UFOs 44 believed in ghosts
25 believed in Bigfoot 10 believed in UFOs and ghosts
8 believed in ghosts and Bigfoot five believed in UFOs and Bigfoot
2 believed in all 3
How many people surveyed believed in at to the lowest degree one of these things?
Endeavor it Now Answers
i. There are several answers: The set of all odd numbers less than 10. The set of all odd numbers. The prepare of all integers. The prepare of all existent numbers.
2. A ⋃ C = {red, orange, yellow, greenish, blueish purple}
Bc ⋂ A = {green, blue}
iii. A ⋃ B ⋂ Cc
4. Starting with the intersection of all three circles, nosotros work our manner out. Since ten people believe in UFOs and Ghosts, and 2 believe in all 3, that leaves 8 that believe in but UFOs and Ghosts. We work our way out, filling in all the regions. Once we have, we can add up all those regions, getting 91 people in the matrimony of all three sets. This leaves 150 – 91 = 59 who believe in none.
Source: https://courses.lumenlearning.com/atd-austincc-mathlibarts/chapter/cardinality/
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