If A is a subset of B and B is a subset of C, then cardinality of A ∪ B ∪ C is equal to

To solve this problem, we need to analyze the cardinality of the union of the sets $A$, $B$, and $C$, given the subset relations $A \subseteq B$ and $B \subseteq C$.

1. ### Break Down the Problem

• We have three sets denoted as $A$, $B$, and $C$.
• We know the relationships between these sets: $A$ is a subset of $B$, and $B$ is a subset of $C$.

2. ### Relevant Concepts

• According to set theory, if $A \subseteq B$ and $B \subseteq C$, then any element in $A$ is also in $B$ and $C$, and any element in $B$ is in $C$.
• The cardinality $|X|$ of a set $X$ is the number of distinct elements in that set.
• Therefore, we can express the cardinalities of the sets based on the given inclusions.

3. ### Analysis and Detail

• Since $A \subseteq B$ and $B \subseteq C$, we can define the cardinalities as follows:
  • Let $|A| = a$
  • Let $|B| = b$ where $b \geq a$
  • Let $|C| = c$ where $c \geq b$
• The union $A \cup B \cup C$ includes all elements from the sets:
  • All elements in $A$ (which are in $B$ and $C$),
  • All elements in $B$ (which are in $C$),
  • All elements in $C$.

Given the subset relations, the unique elements in the union $A \cup B \cup C$ will just be the unique elements in $C$. Therefore, $|A \cup B \cup C| = |C| = c.$

4. ### Verify and Summarize

• Since all elements of $A$ and $B$ are included in $C$, the unique count of elements does not increase in the union.
• Hence, the cardinality of $A \cup B \cup C$ is equivalent to the cardinality of the largest set, which is $C$.

Final Answer

The cardinality of $A \cup B \cup C$ is equal to $|C|$.