intro to sets and venn diagrams
- sets and venn diagrams are used to categorize elements into groups, with particular attention to elements that may belong to multiple groups simultaneously.
- a basic venn diagram includes two overlapping circles within a rectangle, representing two groups and their intersection, alongside those belonging to neither group.
- the interpretation of problem statements is crucial, as slight changes in wording can significantly alter the mathematical approach and solution.
- practice problems demonstrate the process of translating verbal information into a venn diagram, followed by algebraic manipulation to solve for unknown quantities.
- the importance of careful wording interpretation and the utility of sketching venn diagrams for problem-solving are emphasized.
note: here is the text of the practice problem around 6:00 into the video:
at a certain school of 200 students, the students can study french, spanish, both or neither. just as many study neither as study both. one quarter of those who study spanish also study french. the total number who study french is 10 fewer than those who study spanish only. how many students study french only?
q: when should i use a venn diagram and when should i use a double matrix?
a: great question! the answer depends on the characteristics of the population you're trying to consider.
the double matrix
a double matrix is most appropriate when each item/person in our population can be categorized in two different ways. everyone has to fit into two categories.
- each member of the set has quality a or b and quality c or d.
- everything must be in a or b, and everything must be in c or d.
for example, if we have males (a) and females (b) and who are either math majors (c) or not math major (d). we can put everything into a or b and then into c or d. so a double matrix works for this and would be most efficient.
venn diagrams
a venn diagram is best when we have multiple categories and all or some of the categories can overlap. when we have only two categories a and b that can overlap, our choices are:
- a and b both
- a, not b
- b, not a
- neither a nor b
we can use this formula to avoid double-counting the items that are in both a and b: all items = a + b + not a or b - in both a and b.
q: why do you say that 1/4(b + c) = b?
a: let's look at how this is built. first, we know that we are talking about a group called "students who study spanish" and that is b + c (all of the right circle). next, we also are talking about a group that is just the students who study french and spanish, which is just the shared region, b.
so if we know that one quarter of the students who study spanish also study french, we can use these two populations to create a fraction equal to one quarter: b/(b + c) = 1/4. this can be rearranged to (1/4)(b + c) = b like we show in the lesson video. you can multiply the entire equation by 4 to eliminate the fraction, and that is how we got the alternative equation b + c = 4b.