Q.If $A$ and $B$ are finite sets such that $A \subset B$, then $n(A \cup B) =$ ______________.

The key idea here is understanding the definition of a subset and its implication for set union.

1. The condition $A \subset B$ means that every element of set $A$ is also an element of set $B$.
2. The union $A \cup B$ consists of all elements that are in $A$ or in $B$ (or both). …

Let's break down the concept of subsets and unions to understand why this result holds. The core idea here is to visualize what it means for one set to be entirely contained within another, and then consider what happens when you combine them.

Imagine you have a collection of items, say all the fruits in your basket ($B$). Now, suppose you also have a smaller collection of items, specifically all the apples in your basket ($A$). Since all apples are fruits, and these apples are in your basket, it means the set of apples ($A$) is entirely contained within the set of fruits in your basket ($B$). This is precisely what $A \subset B$ means.

When we talk about the union of these two sets, $A \cup B$, we are asking for all items that are either apples or fruits in your basket (or both). Since every apple is already a fruit in your basket, combining the apples with all the fruits in your basket simply gives you all the fruits in your basket. You don't get any new items beyond what was already in $B$.

Let's formalize this step-by-step.

1. Understanding the Subset Condition:

   The notation $A \subset B$ means that every element of set $A$ is also an element of set $B$. In simpler terms, $A$ is entirely contained within $B$.

   For example, if $B = \{1, 2, 3, 4, 5\}$ and $A = \{2, 3\}$, then $A \subset B$ because both $2$ and $3$ are elements of $B$.

2. Visualizing with a Venn Diagram:

   When $A \subset B$, a Venn diagram would show the circle representing set $A$ drawn completely inside the circle representing set $B$. There are no elements in $A$ that are outside of $B$.

   +-----------------------+
   |                       |
   |       B               |
   |    +-----------+      |
   |    |     A     |      |
   |    |           |      |
   |    +-----------+      |
   |                       |
   +-----------------------+
   

3. Defining the Union of Sets:

   The union of two sets, $A \cup B$, is the set containing all elements that are in $A$, or in $B$, or in both.

   Mathematically, $x \in (A \cup B)$ if and only if $x \in A$ or $x \in B$.

4. Applying the Subset Condition to the Union:

   Since $A \subset B$, every element $x$ that belongs to $A$ also belongs to $B$.

   Consider an element $y \in (A \cup B)$. By definition, this means $y \in A$ or $y \in B$.

   • If $y \in B$, then $y$ is an element of $B$.
   • If $y \in A$, then because $A \subset B$, it must also be true that $y \in B$. In both cases, any element in $A \cup B$ must necessarily be an element of $B$. …