Q.Determine whether the following statement is true or false. Justify your answer: For all sets $A$, $B$ and $C$, if $A \subset C$ and $B \subset C$, then $A \cup B \subset C$.

Concept: Subset and union properties

We need to check whether the union of two subsets of $C$ is itself a subset of $C$.

Reasoning:

Assume $A \subset C$ and $B \subset C$. Take any element $x \in A \cup B$.

By definition of union, $x \in A$ or $x \in B$ (or both).

If $x \in A$, then $x \in C$ (since $A \subset C$). If $x \in B$, then $x \in C$ (since $B \subset C$). …

Why this works: Understanding subset relationships

The heart of this problem lies in understanding what it means for one set to be a subset of another. When we write $A \subset C$, we're saying every element of $A$ lives inside $C$. The question asks: if both $A$ and $B$ are contained in $C$, must their union $A \cup B$ also be contained in $C$?

The union $A \cup B$ consists of elements that belong to $A$, or to $B$, or to both. If we know where every element of $A$ goes and where every element of $B$ goes, we can track where every element of their union goes.

Proof by element-chasing

To prove a subset relationship $A \cup B \subset C$, we need to show that every element of $A \cup B$ is also in $C$. This is the standard technique for proving subset claims.

1. Start with an arbitrary element of the union.

   Let $x \in A \cup B$. By the definition of union, this means $x \in A$ or $x \in B$ (or both).

2. Case 1: If $x \in A$.

   We're given that $A \subset C$, which means every element of $A$ is in $C$. Therefore $x \in C$.

3. Case 2: If $x \in B$.

   Similarly, we're given that $B \subset C$, so every element of $B$ is in $C$. Therefore $x \in C$.

4. Conclude the subset relationship.

   In both cases, we've shown that $x \in C$. Since $x$ was an arbitrary element of $A \cup B$, we've proven that every element of $A \cup B$ belongs to $C$. This is precisely what $A \cup B \subset C$ means. …