Introduction to Mathematical Set Theory and Cardinality

Understanding the Complement of a Set

If we have a subset $A$, we are often interested in what is "left over" in the Universal Set. This leads us to the concept of the complement.

The complement of Set A (denoted as $A'$) consists of all the elements that are present in the Universal Set $U$ but are not present in Set A. Essentially, the complement is the "opposite" of the set within the given universe.

Using our previous example:

• $U = {1, 2, 3, 4, 5, 6, 7, 8, 9}$
• $A = {1, 3, 5, 7, 9}$ (the odd numbers)

To find $A'$, we look for every number in $U$ that did not make it into $A$. In this case, the remaining numbers are the even integers: $A' = {2, 4, 6, 8}$

Cardinality: Measuring the Size of Sets

In set theory, we don't just care about which elements are in a set, but also how many elements there are. This measurement is called cardinality.

The cardinality of a set $S$ is denoted by the notation $n(S)$. This represents the total count of distinct elements within that set. Let's apply this to our working example:

1. Cardinality of the Universal Set: To find $n(U)$, we count the elements in ${1, 2, 3, 4, 5, 6, 7, 8, 9}$. $n(U) = 9$
2. Cardinality of Set A: To find $n(A)$, we count the elements in ${1, 3, 5, 7, 9}$. $n(A) = 5$
3. Cardinality of the Complement: To find $n(A')$, we count the elements in ${2, 4, 6, 8}$. $n(A') = 4$

The Fundamental Identity of Cardinality

One of the most important relationships in finite set theory is the mathematical link between a set, its complement, and the universal set. Because the complement $A'$ is defined as everything in $U$ that is not in $A$, the two sets are mutually exclusive (they share no elements) and collectively exhaustive (together they cover the entire universe).

This relationship is expressed through the following fundamental equation: $n(A) + n(A') = n(U)$

This formula tells us that if you add the number of elements in a set to the number of elements in its complement, you must arrive at the total number of elements in the Universal Set.

Worked Example Verification

Let's use the values we calculated earlier to verify this identity:

• $n(A) = 5$
• $n(A') = 4$
• $n(U) = 9$

Plugging these into our equation: $5 + 4 = 9$

The equation holds true, confirming that our division of the Universal Set into Set A and its complement accounts for every element within $U$ without any omissions or overlaps.

Exam Tip: If you are asked to find the cardinality of a complement $n(A')$ and you already know $n(U)$ and $n(A)$, you can simply rearrange the formula: $n(A') = n(U) - n(A)$ This is often a much faster way to solve problems in multiple-choice exams!

Summary Table

To visualize the relationship between these concepts, refer to the table below based on our primary example:

By mastering these definitions and the relationship $n(A) + n(A') = n(U)$, you have built the groundwork necessary for more advanced topics in probability and complex set operations.