Understanding Set Operations and Set Theory: P and Q

This problem from Set Theory involves two sets, P and Q, and asks about their intersection based on a condition involving their union. We are given that for sets P and Q, the Set Union P \cup Q is equal to P.

The given condition illustrating a Set Relationship and Set Equality is: P \cup Q = P.

Analyzing the Set Union Condition: P ∪ Q = P

The Set Union of two sets, P \cup Q, contains all elements that are in P or in Q or in both. The condition P \cup Q = P signifies Set Equality, meaning every element from the combined sets P and Q is already contained within set P itself.

This can only be true if set Q does not contain any element that is not also in set P. In other words, every element of Q must also be an element of P. This is the definition of Subsets.

Thus, the condition P \cup Q = P is equivalent to stating that Q is a subset of P, denoted as Q \subseteq P.

Finding the Set Intersection: P ∩ Q

The Set Intersection of two sets, P \cap Q, consists of all elements that are common to both set P and set Q. Since we know from the given condition that Q \subseteq P, every element in Q is also in P.

Therefore, the elements that are common to both P and Q are exactly the elements that are in Q. The Set Intersection P \cap Q must contain all elements that belong to Q, and no other elements.

Hence, P \cap Q = Q.

Visualizing with a Venn Diagram

Imagine a Venn Diagram representing Set Operations for sets P and Q. The condition P \cup Q = P means the area covering both circles is just the area of the P circle. This graphically shows that the Q circle must be entirely inside the P circle, illustrating that Q is a subset of P.

If the Q circle is inside the P circle, the overlapping region (the intersection, P \cap Q) is simply the Q circle itself. This visualization confirms that P \cap Q = Q when Q is a subset of P.

Conclusion for P ∩ Q

When the union of sets P and Q results in P (i.e., P \cup Q = P), it establishes that Q is a subset of P. Consequently, the elements common to both sets, which form the set intersection P \cap Q, are precisely the elements of Q. Thus, P \cap Q = Q.

This is a key concept in Set Theory involving basic Set Operations and Set Relationships.