NIMCET 2020 — Mathematics PYQ

1. Understand the Subset Relationship:

   The problem states that $A$ is contained within $B$, and $B$ is contained within $C$. Mathematically, this nested relationship is written as:

   $$A\subseteq B\subseteq C$$

2. Analyze the Union ($A\cup B\cup C$):

   The union operation combines all the unique elements from all the given sets. Since $A$ is completely inside $B$, and $B$ is completely inside $C$, all elements of both $A$ and $B$ are already present in $C$.

   Therefore:

   $$A\cup B\cup C=C$$

3. Determine the Cardinality:

   The cardinality represents the total number of elements in a set, denoted as $|\cdot |$. Taking the cardinality of both sides of our previous equation gives:

   $$|A\cup B\cup C|=|C|$$

Thus, the cardinality of the union is exactly equal to the Cardinality of $C$.