Understanding Set Operations in Set Theory

The question asks us to simplify the expression $$(P - Q) \cup (Q - P) \cup (P \cap Q)$$ where P and Q are two sets. This involves combining several fundamental set operations: set difference, intersection, and union. Let's break down each part of the expression based on basic set theory principles.

Breaking Down the Components

Consider two sets, P and Q. We can think of the elements in these sets being divided into different regions:

• P - Q (Set Difference): This represents the set of elements that are in set P but are not in set Q. In terms of a Venn diagram, this is the part of the circle for P that does not overlap with the circle for Q. This is a key concept in understanding set operations.
• Q - P (Set Difference): Similarly, this represents the set of elements that are in set Q but are not in set P. On a Venn diagram, this is the part of the circle for Q that does not overlap with the circle for P. This is another important result of applying the set difference operation.
• P $\cap$ Q (Intersection of Sets): This represents the set of elements that are common to both set P and set Q. On a Venn diagram, this is the overlapping region between the circles for P and Q. This is the result of the intersection of sets operation.

Combining the Components: The Union Operation

The expression asks for the union of sets of these three resulting sets: $$(P - Q) \cup (Q - P) \cup (P \cap Q)$$ The union operation combines all elements from the sets being united. So, we are taking all elements that are either:

• In P but not in Q ($P - Q$)
• OR in Q but not in P ($Q - P$)
• OR in both P and Q ($P \cap Q$)

Let's visualize this using the idea of a Venn diagram for two sets P and Q. The three regions we are considering are mutually exclusive (they have no elements in common):

• Region 1: Elements only in P ($P - Q$)
• Region 2: Elements only in Q ($Q - P$)
• Region 3: Elements in both P and Q ($P \cap Q$)

When we take the union of sets of these three regions, we are including every element that falls into any one of these areas. Together, these three regions cover every single element that is in set P, or in set Q, or in both sets P and Q.

Relating to P $\cup$ Q

By definition, the union of sets P and Q, denoted as $P \cup Q$, is the set of all elements that are in P, or in Q, or in both P and Q. This is exactly what the union of the three regions $(P - Q)$, $(Q - P)$, and $(P \cap Q)$ represents.

Therefore, the expression $$(P - Q) \cup (Q - P) \cup (P \cap Q)$$ simplifies to $P \cup Q$. This is a fundamental result in set theory that shows how the parts of a Venn diagram for two sets relate to the overall union.

Understanding these fundamental set operations is crucial for solving problems in set theory.