Question:
If \(A \subseteq B\) and \(B \subseteq C\), then cardinality of \(A \cup B \cup C\) is equal to:
If \(A \subseteq B\) and \(B \subseteq C\), then cardinality of \(A \cup B \cup C\) is equal to:
Show Hint
Whenever
\[
A\subseteq B\subseteq C,
\]
the union of all sets equals the largest set and the intersection equals the smallest set.
Updated On: Jun 8, 2026
- Cardinality of \(C\)
- Cardinality of \(B\)
- Cardinality of \(A\)
- None of the above
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The Correct Option is A
Solution and Explanation
Concept:
If one set is contained inside another, then taking their union does not create any new elements. The larger set already contains all elements of the smaller set.
Given:
\[
A \subseteq B
\]
and
\[
B \subseteq C
\]
This implies every element of \(A\) belongs to \(B\), and every element of \(B\) belongs to \(C\).
Therefore:
\[
A \subseteq B \subseteq C
\]
Step 1: Find \(A \cup B\).
Since \(A\) is completely contained in \(B\), \[ A \cup B = B \] because union adds no new element.
Step 2: Find \(A \cup B \cup C\).
Using Step 1, \[ A \cup B \cup C = B \cup C \] Again \(B \subseteq C\), therefore \[ B \cup C = C \] Hence \[ A \cup B \cup C = C \]
Step 3: Compare cardinalities.
Since \[ A \cup B \cup C = C \] their cardinalities are equal: \[ |A \cup B \cup C| = |C| \] \[ \boxed{|A\cup B\cup C|=|C|} \]
Step 1: Find \(A \cup B\).
Since \(A\) is completely contained in \(B\), \[ A \cup B = B \] because union adds no new element.
Step 2: Find \(A \cup B \cup C\).
Using Step 1, \[ A \cup B \cup C = B \cup C \] Again \(B \subseteq C\), therefore \[ B \cup C = C \] Hence \[ A \cup B \cup C = C \]
Step 3: Compare cardinalities.
Since \[ A \cup B \cup C = C \] their cardinalities are equal: \[ |A \cup B \cup C| = |C| \] \[ \boxed{|A\cup B\cup C|=|C|} \]
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