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# Set Theory

## Surveys

Problems involving sets of people (or other objects) sometimes require analyzing known information about certain subsets to obtain cardinal numbers of other subsets. The "known information" is often obtained by administering a survey.

Example: Suppose that a group of 150 people were questioned about particular social media platforms that they used.

90 used Instagram, 62 used TikTok, and 38 used Facebook.

42 used Instagram and TikTok, 20 used TikTok and Facebook, 25 used Instagram and Facebook, and 18 used all three

How can we analyze this survey using a Venn Diagram?

Construct a Venn Diagram. Let I = set of those who used Instagram, T = set of those who used TikTok, and F = set of those who used Facebook.

From the final Venn Diagram, we can conclude

41 people only use Instagram, 18 people only use TikTok, 11 people only use Facebook, 29 people do not use any platforms.

Other conclusions include 7 people use Instagram and Facebook, this does not include people who use all 3 or people that just use one of the platforms. In set notation the region where 7 is located is represented by $$n\left(I \cap F \right) - n\left(I \cap T \cap F \right)$$

Cardinal Number Formula Relating the Union and Intersection of two sets. You will see this again as not mutually exclusive sets.

$n\left(A \cup B\right) = n\left(A \right) + n\left(B\right) - n\left(A \cap B\right)$