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At the heart of set theory lies the concept of subset and superset relationships. When we say 'subset,' we are referring to a set whose elements are all contained within another set. Imagine a bag of red marbles within a larger bag of various colored marbles; the red marbles form a subset of the larger marble collection.

Formally, if every element in set A is also in set B, we say that A is a subset of B, and write it as \( A \subseteq B \). For example, if set A is {1, 2} and B is {1, 2, 3, 4}, A is a subset of B because all elements of A are in B.

A superset, on the other hand, is the larger set that contains all elements of its subsets. Continuing with our previous example, set B is a superset of set A, because it contains all elements of A and possibly more. We can denote this relationship as \( B \supseteq A \).

A couple of key points to keep in mind are:

• If set A is a subset of B and B is a subset of A, then A and B are actually the same set, known as 'equal sets'.
• Every set is a subset of itself, which means \( A \subseteq A \)

The union of two sets is a fundamental operation that combines all the elements from both sets, without duplication. In our analogy, if you have one bag of red marbles and another of blue, then the union of these two bags will give you a mixed bag of red and blue marbles.

The union is symbolically represented by the cup symbol \( \cup \), and mathematically, it's defined as \( A \cup B = \{ x : x \in A \text{ or } x \in B \} \). This definition may sound complex, but it simply means that the union set consists of all elements that are in A, in B, or in both A and B.

For instance, if set A is {1, 2, 3} and set B is {3, 4, 5}, then the union \( A \cup B \) is the set {1, 2, 3, 4, 5}. Notice that '3' is not listed twice, because in set theory, we do not consider repetitions.

The union operation is associative and commutative, which means that the order in which sets are united doesn't change the result \( ((A \cup B) \cup C = A \cup (B \cup C)) \) and \( A \cup B = B \cup A \).

The intersection of sets refers to a new set made up of the common elements that the sets share. This is like finding all the people who appear on both the guest lists of two separate parties.

For the intersection, we use the cap symbol \( \cap \), and it's defined as \( A \cap B = \{ x : x \in A \text{ and } x \in B \} \). The elements of \( A \cap B \) must be in both A and B.

If we stick with our example sets A = {1, 2, 3} and B = {3, 4, 5}, their intersection \( A \cap B \) is the set {3} since that is the only element they have in common. An important note is that if two sets have no elements in common, their intersection is an empty set, denoted by \( \emptyset \).

Just as with the union operation, the intersection is commutative and associative, meaning \( A \cap B = B \cap A \) and \( (A \cap B) \cap C = A \cap (B \cap C) \).