Question

If \(A \subseteq B\) and \(B \subseteq C\), then cardinality of \(A \cup B \cup C\) is equal to:

• A. Cardinality of \(C\)
• B. Cardinality of \(B\)
• C. Cardinality of \(A\)
• D. None of the above

Hint:

Whenever \[ A\subseteq B\subseteq C, \] the union of all sets equals the largest set and the intersection equals the smallest set.

Answer 1

The Correct Option is A

Step 1: Read what is given.
We are told $A \subseteq B$ and $B \subseteq C$. This means every element of $A$ is inside $B$, and every element of $B$ is inside $C$.

Step 2: Chain the subsets.
Putting both facts together, we get $A \subseteq B \subseteq C$. So $C$ is the biggest set and it already holds everything.

Step 3: Simplify $A \cup B$ first.
Since $A$ sits fully inside $B$, joining them adds nothing new.
\[ A \cup B = B \]
Step 4: Now bring in $C$.
So $A \cup B \cup C = B \cup C$.

Step 5: Simplify $B \cup C$.
Since $B$ sits fully inside $C$, joining them adds nothing new either.
\[ B \cup C = C \]
So $A \cup B \cup C = C$.

Step 6: Compare the sizes.
If two sets are exactly the same set, they have the same number of elements. So the cardinality of $A \cup B \cup C$ equals the cardinality of $C$.
\[ \boxed{|A \cup B \cup C| = |C|} \]