Q.Given the sets $A = \{1, 3, 5\}$, $B = \{2, 4, 6\}$ and $C = \{0, 2, 4, 6, 8\}$. Then the universal set of all the three sets $A$, $B$ and $C$ can be ______________.

The universal set for a given collection of sets is a set that contains all elements of all the sets in that collection. To find such a set, we must combine all unique elements from the given sets.

1. The key idea is that a universal set must encompass every element from the sets it describes.
2. Therefore, we need to find the union of the sets $A$, $B$, and $C$.
3. First, find the union of $A$ and $B$: $A \cup B = \{1, 3, 5\} \cup \{2, 4, 6\} = \{1, 2, 3, 4, 5, 6\}$ …

In set theory, a universal set (often denoted by $U$) is a set that contains all elements under consideration for a particular problem or context. It acts as a "super-set" from which all other sets in that context are drawn. For a collection of sets, say $A$, $B$, and $C$, a universal set must contain every single element that is present in $A$, or in $B$, or in $C$. It's like defining the entire scope of elements you're working with.

There isn't just one unique universal set for any given collection of sets; any set that contains all the elements of the given sets can serve as a universal set. However, the most common and often implied universal set, especially when no other context is provided, is the smallest possible set that contains all elements from the given sets. This smallest set is precisely the union of all the given sets.

Let's find a universal set for $A$, $B$, and $C$.

1. Identify the elements in each given set.

   We are given the following sets:

   • $A = \{1, 3, 5\}$
   • $B = \{2, 4, 6\}$
   • $C = \{0, 2, 4, 6, 8\}$

2. Understand the requirement for a universal set.

   For a set to be a universal set for $A$, $B$, and $C$, it must contain every element that appears in $A$, every element that appears in $B$, and every element that appears in $C$. In other words, it must contain all unique elements from all three sets combined.

3. Combine all unique elements using the union operation.

   The operation that combines all unique elements from multiple sets is the union. We need to find $A \cup B \cup C$.

   The union of sets $X$ and $Y$, denoted $X \cup Y$, is the set containing all elements that are in $X$, or in $Y$, or in both.

   First, let's find the union of $A$ and $B$:

   $A \cup B = \{1, 3, 5\} \cup \{2, 4, 6\}$

   Since there are no common elements between $A$ and $B$, their union simply lists all elements from both sets:

   $A \cup B = \{1, 2, 3, 4, 5, 6\}$ …