So am I. Aren’t we all? Our brain likes it that way.
Let us activate our visual cortex to learn more about Sets. Imagine that you are taking a headcount for high school or college intramurals. You have players who are interested in basketball and players who are interested in soccer. Some of them are interested in participating in both basketball and soccer.
Let us ask them to organize themselves into the basketball and soccer teams. A simple visual of this would look like:
Let us call our basketball players Team Basket (B). Let us call our soccer players Team Soccer (S). Let us call all of them Team Intramural (I).
If Team Intramural is the set of all players interested in participating in sports, Team Basket and Team Soccer are subsets of Team Intramural. All the players in Team Basket and Team Soccer also belong to Team Intramural.
All players who do not play soccer are the complement of Team Soccer. The complement of a set is used to denote everything but the set.
If you want to know how many players are interested in basketball or soccer, you can visualize the entire space that includes Team Basket or Team Soccer. The union of these two teams or set.
If you want to know how many players are interested in basketball and soccer, you can visualize the space that includes players that like to participate in both. The individuals who intersect the two teams.
If you want to take a count of players interested in only one sport but not both, you can visualize a space that includes players that like basketball or players that like soccer, but the players that like both will miss out. This grouping is also called symmetric difference, or the Union – Intersection.
Now imagine another class of students approached you inquiring about field hockey. Let us call this new team, Team Hockey. Team Hockey will become a subset of Team Intramural. However, the players of Team Hockey and the players of Team Soccer do not have anything in common. They are mutually exclusive. They are disjoint. They don’t cross lines — their intersection is 0.
Next week, we will learn how to use real data in RStudio, how to categorize the data into sets, and how to check for intersections, unions, and exclusiveness. In the meantime, try to visualize how our players and teams fit these other Set Properties.
If the union of all sets makes up the entire space, then the sets are collectively exhaustive. Our Team Soccer, Team Basketball, and Team Hockey are collectively Team Intramural (Exhaustive).
I finally think my students got the idea of mutually exclusive and collectively exhaustive sets when I told them that the mid terms are mutually exclusive, but the final exam is collectively exhaustive !! They must have visualized the final.
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