Cardinality
Learning Outcomes
- Describe memberships of sets, including the empty set, using proper notation, and decide whether given items are members and determine the cardinality of a given set.
- Describe the relations between sets regarding membership, equality, subset, and proper subset, using proper notation.
- Perform the operations of union, intersection, complement, and difference on sets using proper notation.
- Be able to draw and interpret Venn diagrams of set relations and operations and use Venn diagrams to solve problems.
- Recognize when set theory is applicable to real-life situations, solve real-life problems, and communicate real-life problems and solutions to others.
Cardinality
The number of elements in a set is the cardinality of that set. The cardinality of the set A is often notated as |A| or n(A)Exercises
Let A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8}. What is the cardinality of B? A ⋃ B, A ⋂ B?Answer: The cardinality of B is 4, since there are 4 elements in the set. The cardinality of A ⋃ B is 7, since A ⋃ B = {1, 2, 3, 4, 5, 6, 8}, which contains 7 elements. The cardinality of A ⋂ B is 3, since A ⋂ B = {2, 4, 6}, which contains 3 elements.
Exercises
What is the cardinality of P = the set of English names for the months of the year?Answer: 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 set. This is common in surveying.
Exercises
A survey asks 200 people “What beverage do you drink in the morning”, and offers choices:- Tea only
- Coffee only
- Both coffee and tea
Answer: This question can most easily be answered by creating a Venn diagram. We can see that we can find the people who drink tea by adding those who drink only tea to those who drink both: 60 people. We can also see that those who drink neither are those not contained in the any of the three other groupings, so we can count those by subtracting from the cardinality of the universal set, 200. 200 – 20 – 80 – 40 = 60 people who drink neither.
Example
A survey asks: Which online services have you used in the last month:- Have used both
Answer: Let T be the set of all people who have used Twitter, and F be the set of all people who have used Facebook. Notice that while the cardinality of F is 70% and the cardinality of T is 40%, the cardinality of F ⋃ T is not simply 70% + 40%, since that would count those who use both services twice. To find 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) n(F ⋃ T) = 70% + 40% – 20% = 90%
Example
Now, to find how many people have not used either service, we’re looking for the cardinality of (F ⋃ T)c .Answer: Since the universal set contains 100% of people and the cardinality of F ⋃ T = 90%, the cardinality of (F ⋃ T)c must be the other 10%.
Cardinality properties
- n(A ⋃ B) = n(A) + n(B) – n(A ⋂ B)
- n(Ac) = n(U) – n(A)
n(A ⋂ B) = n(A) + n(B) – n(A ⋃ B)
Example
Fifty students were surveyed, and asked if they were taking a social science (SS), humanities (HM) or a natural science (NS) course the next quarter. 21 were taking a SS course 26 were taking a HM course 19 were taking a NS course 9 were taking SS and HM 7 were taking SS and NS 10 were taking HM and NS 3 were taking all three 7 were taking none How many students are only taking a SS course?Answer: It might help to look at a Venn diagram. From the given data, we know that there are 3 students in region e and 7 students in region h. Since 7 students were taking a SS and NS course, we know that n(d) + n(e) = 7. Since we know there are 3 students in region 3, there must be 7 – 3 = 4 students in region d. Similarly, since there are 10 students taking HM and NS, which includes regions e and f, there must be 10 – 3 = 7 students in region f. Since 9 students were taking SS and HM, there must be 9 – 3 = 6 students in region b. Now, we know that 21 students were taking a SS course. 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 – 6 – 4 – 3 = 8 students are in region a. 8 students are taking only a SS course.
Try It
One hundred fifty 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 5 believed in UFOs and Bigfoot 2 believed in all three How many people surveyed believed in at least one of these things?Answer: 1. There are several answers: The set of all odd numbers less than 10. The set of all odd numbers. The set of all integers. The set of all real numbers. 2. A ⋃ C = {red, orange, yellow, green, blue purple} Bc ⋂ A = {green, blue} 3. A ⋃ B ⋂ Cc 4. Starting with the intersection of all three circles, we work our way out. Since 10 people believe in UFOs and Ghosts, and 2 believe in all three, that leaves 8 that believe in only 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 union of all three sets. This leaves 150 – 91 = 59 who believe in none.
Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- Math in Society. Authored by: Open Textbook Store, Transition Math Project, and the Open Course Library. Located at: http://www.opentextbookstore.com/mathinsociety/. License: CC BY-SA: Attribution-ShareAlike.
- Sets: cardinality. Authored by: David Lippman. License: CC BY: Attribution.
- Question ID 125872, 109842, 125878. Authored by: Bohart,Jenifer, mb Meacham,William. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.