Sets: Terminologies, Notations, and Operations

As a preparation for more posts on probability, statistics, permutations and combinations, we familiarized ourselves last week with the different terminologies and notations of probability.  We continue in this post by studying set terminologies, notations, and operations. Note that this is also the third post in the Set Primer Series; the first and second are Introduction to Sets and  Subset: a set contained in a set.

Universal Set

The universal set is the set that contains all the elements under discussion. If we talk about the letters  in the English alphabet, then the universal set contains all the 26 letters. In set theory, universal set is usually denoted by U.

In the following discussion, we let U be the set of integers, E be the set of even integers, and O be the set of odd integers.  The following are the common operations on sets.

Intersection

If sets A and B have elements in common they form a set written as A \cap B. This is the intersection of A and B.

Intersection of Sets

Example: If we let A = \{1, 2, 3, 4, 5\} and B = \{2, 4, 6\} then A \cap B = \{2,4\}. » Read more

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.

sets-and-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

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