Month: February 2017

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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We begin our quest with the idea of classifying things. Whether it is Aristotle grouping animals into those living in water and those living on land, or you and me grouping our daily activities into those best done in the morning and those best done in the evening, we are all obsessed with putting things in order – into blocks — groups — SETS.

Math puritans can start with Georg Cantor’s Set Theory. Others can think of SET as a collection of distinct elements.

The fruit basket in your house is a set of fruits consisting of apples, bananas, and grapes – {apple, banana, grapes}.

After work, you can visit a local hangout place where you find a set of people interested in alcohol or food or both.

The vowels in English alphabets are a set {a, e, i, o, u}. The English alphabets are a set {a, b, c, …, x, y, z}.

The outcomes of a coin toss are a set {Head, Tail}.

In the game of Monopoly, you move by the outcomes of the dice. These outcomes are a set of number combinations – {(1,1), (1,2) … (6,6)}.

We all tried rolling a double to get out of jail. Did you know that the odds of getting out of jail by rolling a double are only 16.6%?

There are six possible doubles (see the red background combinations along the diagonal – {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}) when you roll two dice. The entire set of possible combinations are 36. The odds are 6/36. Maybe you should have paid the $50 to get out of jail immediately.

Think about sets and possible outcomes in whatever you do this week — Happy President’s Day.

Speaking of Presidents, you should have already imagined a Set of

 

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Data is the key to understanding patterns, learning about behaviors, testing your theories, and supporting your arguments. It provides an entry point to get a general idea about anything. Make a commitment to yourself that you will think about data when you see something. Here, I provide some common situations to prime the pump.

• Preferred Coffee: I am writing this post from a coffee shop. My favorite coffee is espresso. I am curious about what others visiting this shop prefer. I can either sit all day and watch what they buy (which will get the manager suspicious) or ask the cashier about how many people purchased espresso or other coffee (tell them it’s a scientific experiment first!).

• Cars and Tolls: We all waited in line to pay the toll to use the bridge. Friends from the tri-state area are thinking EZPASS. Yes, next time you pass a toll, think about how many cars pass the toll in a day. We can use this data to understand how many people use the bridge and how much revenue it generates.

• Fitbit: Look at your Fitbit or smartphone health app. Tell me how many hours your sleep on average.

• You get the point.

 

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You can also follow me on Twitter @realDevineni for updates on new lessons.

Uncertainties surround us. Understanding where they arise from and recognizing their extent is fundamental to improving our knowledge about anything. This website is all about analyzing the data to interpret the uncertainties. It will contain series of blog posts attempting to explain key ideas in bite sizes. Post your questions in the comments bar. Happy learning!