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Set theory is a branch of mathematical logic that deals with collections of objects, which are called sets. A set can be made up of anything: numbers, letters, symbols, or other unique items. Studying sets helps us understand how to group items and how these groups (or sets) relate to each other. In set theory, elements of a set can be anything, as long as they follow certain rules of membership. Sets are typically denoted by capital letters such as 'A', 'B', or 'C', and their elements might be written in curly braces.

For example, if we have a set of numbers between 1 and 5, we could write it as \({1, 2, 3, 4, 5}\). If another set contains the first three numbers, we would write it as \(\{1, 2, 3\}\). Set theory provides a language to discuss collections of objects and has applications throughout mathematics and science.

A subset is a set where all its elements are contained within another set. Suppose we have sets A and B; if every element of set A is also in set B, A is said to be a subset of B, denoted as \( A \subseteq B \). In our exercise, set A is a subset of set B because all elements from set A are part of set B. However, they are not equal sets, given \( A eq B \).

There are different types of subsets:

• Proper Subset: A proper subset is a subset that is not identical to its parent set. In this case, A is a proper subset of B if element(s) exist in B that are not in A. This distinction matters in our exercise since A must be a proper subset of B, with \( A \subset B \).
• Improper Subset: An improper subset is just another way of talking about the set itself; hence, every set is an improper subset of itself.

Understanding subsets helps mathematicians to define limitations and extents of different relationships between sets.

The union of two sets combines all their elements into a single set, removing any duplicate elements. Mathematically, the union operation is denoted by the symbol \( \cup \). If you have two sets, A and B, then the union \( A \cup B \) results in a set that contains every element from both sets.

For example, if \( A = \{1, 2, 3\} \, and \, B = \{3, 4, 5\}\), then \( A \cup B = \{1, 2, 3, 4, 5\} \). It's a straightforward operation but critical in understanding how sets can interact and combine. In our original exercise, the union of A and B is equal to B (i.e., \( A \cup B = B \)). This informs us that every element in A is already included in B, reinforcing A's status as a subset of B.

Mathematical illustration is a powerful tool for visualizing relationships between sets through diagrams and drawings. One of the most common diagrams used in set theory is the Venn diagram, which visually represents sets as overlapping circles.

Each circle represents a set, and the overlapping areas demonstrate common elements shared by the sets. For the exercise at hand, a Venn diagram illustrates the relationship where set A is completely enclosed within set B. This indicates that all elements of A are in B, but not all elements of B are in A.

• Draw two circles: a bigger one representing set B and a smaller one completely inside it representing set A.
• This visual method immediately shows that A is a subset of B.

Venn diagrams help us quickly understand complex relationships and simplify set theory concepts by making abstract ideas concrete and visually intuitive.