Set Properties Explained: A ⊆ B ⊆ C

This question explores the relationship between sets and their cardinalities when subset properties are involved.

We are given two conditions:

• A is a subset of B, which means every element in set A is also present in set B. Mathematically, this is written as $A \subseteq B$.
• B is a subset of C, meaning every element in set B is also present in set C. Mathematically, this is written as $B \subseteq C$.

From these two conditions, we can deduce a relationship between A and C. Since all elements of A are in B, and all elements of B are in C, it logically follows that all elements of A must also be in C. Therefore, A is also a subset of C ($A \subseteq C$).

Understanding Set Union: A ∪ B ∪ C

The union of sets, denoted by the symbol $\cup$, combines all the unique elements from the sets involved.

The expression $A \cup B \cup C$ represents the set containing all elements that belong to A, or to B, or to C, or to any combination of these sets.

Determining the Union with Subset Conditions

Given that $A \subseteq B$ and $B \subseteq C$:

• Because $A \subseteq B$, all elements of A are already included in B.
• Because $B \subseteq C$, all elements of B are already included in C.

Consequently, when we take the union $A \cup B \cup C$:

• The elements of A are already accounted for in B.
• The elements of B are already accounted for in C.
• Therefore, the union of A, B, and C consists solely of all the elements present in C. No elements outside of C can be part of this union because A and B are themselves subsets of C.

This leads us to the conclusion that the union of these three sets is simply the largest set, C: $A \cup B \cup C = C$.

Cardinality of the Union Set

Cardinality refers to the number of elements in a set. Let $|X|$ denote the cardinality of a set X.

Since we established that $A \cup B \cup C = C$, their cardinalities must be equal.

Therefore, $|A \cup B \cup C| = |C|$.

This means the cardinality of the union of A, B, and C is equal to the cardinality of C.