91Ó°ÊÓ

The union of sets is a fundamental concept in set theory. It involves combining the elements of two or more sets to form a new set that includes every distinct element from each of them. When we talk about the union of sets, we think about it as combining all items while keeping each one unique.

For example, let's consider the sets:

• Set B: \( \{1, 2, 3\} \)
• Set C: \( \{2, 3, 4, 5, 6\} \)

The union of B and C, denoted by \( B \cup C \), would include every number that appears in either of the two sets. This means:
\[ B \cup C = \{1, 2, 3, 4, 5, 6\} \]

We capture each element that appears in B or C. Note that any repeat numbers only appear once in the result. This is important because sets do not have duplicate elements.

The complement of a set refers to all the elements that are not in the given set but are in a universal set. The universal set is a collection of all possible elements we are considering for a particular discussion. The complement is denoted as \( A^c \) (or sometimes \( \overline{A} \)).

Let's use the universal set \( U = \{1, 2, 3, 4, 5, 6, 7\} \). Suppose we have a set \( A = \{1, 3, 5, 7\} \). The complement of set A, \( A' \), is every element that is in U but not in A. In this example:

• Complement of A: \( A' = \{2, 4, 6\} \)

When working with complements, it's crucial to always know what your universal set is. Without defining it, complements can be confusing as it defines the boundary of potential elements.

The intersection of sets is the operation that finds elements common to all sets involved. When we calculate the intersection, we are interested in what overlaps or what elements are shared between the sets. This is expressed as \( A \cap B \).

In the solution, we find the intersection between the complement of \( B \cup C \) and set A. The sets are:

• \( (B \cup C)' = \{7\} \)
• \( A = \{1, 3, 5, 7\} \)

The intersection, \( (B \cup C)' \cap A \), is simply:
\[ (B \cup C)' \cap A = \{7\} \]
This means that the number 7 is the only common element between the two sets. Intersection is fundamental in determining shared characteristics or agreements across sets.

The universal set is a collection that contains all possible elements for a specific discussion or problem. It provides context and boundaries for talking about other sets and their relationships, like complements or intersections.

Consider the universal set represented in the exercise:

• \( U = \{1, 2, 3, 4, 5, 6, 7\} \)

The universal set includes every number examined in this problem. All other sets, like A, B, and C, draw their elements from this universal set.

When discussing complements and unions, the universal set is crucial because it frames what's available and what might be excluded when forming new sets through operations such as \( A' \) (complement) or \( A \cup B \) (union). Understanding the universal set ensures clear understandings of set operations’ outcomes.