Understanding Set Union and Cardinality

The question asks about the minimum number of elements possible in the union of two sets, set A and set B. We are given the number of elements in each set:

• Set A contains 3 elements. We can write this as $|A| = 3$.
• Set B contains 6 elements. We can write this as $|B| = 6$.

The union of two sets, denoted as $A \cup B$, is the set containing all elements that are in set A, or in set B, or in both. The number of elements in the union of two sets is given by the formula:

$$|A \cup B| = |A| + |B| - |A \cap B|$$

Here, $|A \cap B|$ represents the number of elements in the intersection of set A and set B. The intersection, $A \cap B$, is the set containing only the elements that are common to both set A and set B.

Minimizing the Number of Elements in the Union

We want to find the minimum possible value for $|A \cup B|$. Looking at the formula $|A \cup B| = |A| + |B| - |A \cap B|$, we can see that to minimize $|A \cup B|$, we need to maximize $|A \cap B|$. This is because $|A|$ and $|B|$ are fixed values (3 and 6 respectively), and we are subtracting $|A \cap B|$. A larger $|A \cap B|$ will result in a smaller $|A \cup B|$.

Finding the Maximum Intersection

The intersection of two sets, $A \cap B$, can contain at most the number of elements present in the smaller of the two sets. This is because the elements in the intersection must be present in both sets. If an element is not in the smaller set, it cannot be in the intersection.

In this case, set A has 3 elements and set B has 6 elements. The smaller set is A, with 3 elements. Therefore, the maximum possible number of elements in the intersection $|A \cap B|$ is 3. This occurs when all elements of set A are also elements of set B, meaning set A is a subset of set B ($A \subseteq B$).

Calculating the Minimum Union Size

Now we use the formula for the union size, substituting the maximum possible intersection value:

$$|A \cup B|_{\text{min}} = |A| + |B| - |A \cap B|_{\text{max}}$$

Substitute the given values $|A|=3$, $|B|=6$, and $|A \cap B|_{\text{max}}=3$:

$$|A \cup B|_{\text{min}} = 3 + 6 - 3$$

$$|A \cup B|_{\text{min}} = 9 - 3$$

$$|A \cup B|_{\text{min}} = 6$$

Thus, the minimum number of elements that the union $(A \cup B)$ can have is 6. This minimum occurs when set A is a subset of set B.

Summary of Calculation

• Given $|A|=3$ and $|B|=6$.
• Formula for union: $|A \cup B| = |A| + |B| - |A \cap B|$.
• To minimize $|A \cup B|$, maximize $|A \cap B|$.
• Maximum $|A \cap B|$ is $\min(|A|, |B|) = \min(3, 6) = 3$.
• Minimum $|A \cup B| = 3 + 6 - 3 = 6$.

Revision Table: Set Theory Concepts

Additional Information: Union and Intersection Bounds

Understanding the bounds for the number of elements in the union and intersection of two sets is important:

• Intersection ($|A \cap B|$):
  • The minimum number of elements in the intersection is 0. This happens when the sets A and B are disjoint (they have no elements in common).
  • The maximum number of elements in the intersection is the size of the smaller set, i.e., $\min(|A|, |B|)$. This happens when the smaller set is a subset of the larger set.
  • So, $0 \le |A \cap B| \le \min(|A|, |B|)$.
• Union ($|A \cup B|$):
  • Using the formula $|A \cup B| = |A| + |B| - |A \cap B|$, we can find the bounds for the union.
  • To find the minimum union size, we use the maximum intersection size: $|A \cup B|_{\text{min}} = |A| + |B| - \min(|A|, |B|) = \max(|A|, |B|)$. This occurs when one set is a subset of the other. In our case, $\max(3, 6) = 6$.
  • To find the maximum union size, we use the minimum intersection size (0): $|A \cup B|_{\text{max}} = |A| + |B| - 0 = |A| + |B|$. This occurs when the sets are disjoint. In our case, $3 + 6 = 9$.
  • So, $\max(|A|, |B|) \le |A \cup B| \le |A| + |B|$. In our case, $6 \le |A \cup B| \le 9$.

The question specifically asked for the minimum number of elements in the union, which we found to be 6.