## Intro to Sets and Venn Diagrams

- Introduction to sets and Venn Diagrams, highlighting their utility in representing complex classifications and overlaps between groups.
- Explanation of basic Venn Diagram components and how they represent members of one group, both groups, or neither group.
- Detailed walkthrough of solving problems using Venn Diagrams, emphasizing the importance of careful interpretation of the problem's wording.
- Illustration of mathematical strategies to solve for unknowns in Venn Diagram problems, including algebraic manipulation and logical deduction.
- Advice on sketching Venn Diagrams for problem-solving and the significance of distinguishing between 'all in' and 'all in only' within problem statements.

__Text of the French/Spanish question in the lesson__

At a certain school of 200 student, 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.