Milkshakes and Power Sets

In Milkshakes, Beads, and Pascal’s Triangles, we have talked about  a systematic way of choosing a combination of objects from a larger number of objects. Let us recall the problem in the said post.

Issa went to a shake kiosk and want to buy a milkshake. The shake vendor told her that she can choose plain milk, or she can choose to combine any number of flavors in any way she wants. There are four flavors to choose from: Apple, Banana, Chico, and Durian. How many possible combination of flavors can Issa make? 

In the problem, Issa can choose any number of flavors and any combination. She can choose plain milk, choose one flavor at a time, two flavors at a time, three flavors at a time, or four flavors at a time as shown in the table below (click the table to enlarge).

Notice that in writing the list, we have exhausted the number of subsets in a set with four elements.  If we let a, b, c, and d stand for avocado, banana, chico, and durian, let them be members or a set and use the set notation, we can write the subsets as follows: » Read more

Subset: a set contained in a set

Two weeks ago, we have talked about the basics of sets.  In this post, we are going to talk about subsets.


If you understand what this means, then you have a notion of a subset.

In mathematics, if A and B are sets, we say that A is a subset of B, denoted by A \subseteq B, if all elements of A are also elements of B. The easiest way to illustrate this is through a Venn diagram as shown on the right.  In the Venn diagram, set A is within set B. Therefore, all elements of set A are also elements of B.

Example 1

LetB be set of all letters in the English alphabet and A be the set of vowel letters. It is clear that A \subseteq B since {aeiou} are elements A and also are elements of B. » Read more

Sets: A Gentle Introduction

There are people who are fond of buying things in sets. For example, a pen collector may collect pens of different colors.   Other people are obsessed with brands.   For instance, Apple fans may buy the following an iPhone, iPod, iPad, and an iMac. If you are fond of reading Dan Brown, you probably have read or own some of the following books: The Lost SymbolDeception PointDigital Fortress, and The Da Vinci Code, and others.


In mathematics, a set is not very different.  A set is can be described as a collection of objects with common characteristics. These objects are called elementsGiven a set and an object, it is clear if the object is an element of a set or not.  If we let D be a set of Dan Brown novels, then D = { The Lost SymbolDeception PointDigital Fortress, The Da Vinci Code}. Also, if we let V be the set of vowel letters in the English alphabet, then V = {a, e, i, o, u}, and if we let R be the set of polygons whose number sides is less than 9, then the set would include the polygons in the figure below. » Read more

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