Complement of a Set

Given a set $ B $, which is a subset of a larger set $ A $, the complement of $ B $ relative to $ A $ is the set of elements that belong to $ A $ but not to $ B $. In symbolic notation, this is expressed as: \[C_A(B) = A - B\] or, more concisely: \[ \overline{B}_A = A - B \]
example of a complement set

In set theory, the concept of a set complement is fundamental for describing everything that is excluded from a given set.

But what does the complement of a set actually mean? Let's break it down with a clear definition and some concrete examples.

If we consider the universal set $ U $, which contains all the elements relevant to the context, the complement of a set $ A $ relative to $ U $ is the collection of elements that are in $ U $ but not in $ A $. This is denoted by $ A^C $ or $ \overline{A} $, meaning: \[ A^C = U - A \] In other words, the complement of a set consists of everything that lies outside of it within the given universal set.

Practical Examples

To get a clearer picture, let's consider some real-world examples.

Imagine we have a universal set $ U $ representing all the students in a school. Suppose there are 100 students in total:

$$ U = 100 $$

Now, let’s define a subset $ A $, which consists of students who participate in sports.

For instance, if 60 students are involved in sports, we have:

$$ A = 60 $$

To find the complement of $ A $ with respect to $ U $, we determine the number of students who do not participate in sports:

$$ A^C = U - A = 100 - 60 = 40 $$

Thus, the complement of $ A $ is the set of 40 students who do not engage in any sports activities.

This example illustrates how the complement of a set helps define what falls outside a particular category in a structured way.