Given the number of elements in two sets $A$ and $B$, we can answer each part as follows:

(a) The maximum possible number of elements in $A \cup B$ (union of sets):

The union of two sets $A \cup B$ contains all elements from both sets. The maximum occurs when $A$ and $B$ have no common elements (i.e., they are disjoint). In this case:

$|A \cup B| = |A| + |B| = 41 + 22 = 63$

So, the maximum number of elements in $A \cup B$ is 63.

(b) The minimum possible number of elements in $A \cup B$:

The minimum occurs when $A$ and $B$ overlap completely (i.e., $A \subseteq B$ or $B \subseteq A$). In this case:

$|A \cup B| = \max(|A|, |B|) = \max(41, 22) = 41$

So, the minimum number of elements in $A \cup B$ is 41.

(c) The maximum possible number of elements in $A \cap B$ (intersection of sets):

The maximum occurs when $A$ and $B$ completely overlap (i.e., one set is a subset of the other). In this case, the intersection contains all elements of the smaller set:

$|A \cap B| = \min(|A|, |B|) = \min(41, 22) = 22$

So, the maximum number of elements in $A \cap B$ is 22.

(d) The minimum possible number of elements in $A \cap B$:

The minimum occurs when $A$ and $B$ have no common elements (i.e., they are disjoint). In this case:

$|A \cap B| = 0$

So, the minimum number of elements in $A \cap B$ is 0.

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Related Questions:

1. If $A$ and $B$ have 5 common elements, how many elements are in $A \cup B$?
2. If $A$ and $B$ are disjoint, how does that affect $|A \cup B|$ and $|A \cap B|$?
3. How many subsets can be created from set $A$?
4. What is the number of elements in $A \setminus B$ (the difference between sets $A$ and $B$)?
5. How does the inclusion-exclusion principle apply to calculating $|A \cup B|$?

Tip:

To determine the number of elements in unions or intersections of sets, using Venn diagrams can help visualize the relationships clearly.