Let's analyze and solve the given problem step-by-step:

Provided Information

1. Universal Set, $U$: $\{1, 2, 3, 4, 5, 6\}$
2. Set $A$: $\{1, 2, 3, 4\}$
3. Set $B$: $\{3, 5, 6\}$
4. Set $C$: $\{3, 4, 5, 6\}$

The questions to solve are:

1. $(i)$ Find $(A \cup B)'$.
2. $(ii)$ Find $A' \cap B'$.
3. $(iii)$ What do you notice?
4. $(iv)$ Prove that $(A \cap B)' = A' \cup B'$.
5. $(v)$ Find $A \Delta B$.

Solutions

1. (i) $(A \cup B)'$:

   • $A \cup B$ represents the union of sets $A$ and $B$, which includes all elements in either $A$ or $B$. $A \cup B = \{1, 2, 3, 4, 5, 6\}$
   • The complement, $(A \cup B)'$, includes all elements in $U$ but not in $A \cup B$. $(A \cup B)' = U - (A \cup B) = \{\}$

2. (ii) $A' \cap B'$:

   • $A'$ is the complement of $A$, meaning all elements in $U$ but not in $A$. $A' = \{5, 6\}$
   • $B'$ is the complement of $B$, meaning all elements in $U$ but not in $B$. $B' = \{1, 2, 4\}$
   • The intersection, $A' \cap B'$, includes all elements that are in both $A'$ and $B'$. $A' \cap B' = \{\}$

3. (iii) What do you notice?

   • Notice that $(A \cup B)' = A' \cap B'$. This is a standard property in set theory, known as De Morgan's Law.

4. (iv) Prove that $(A \cap B)' = A' \cup B'$:

   • By De Morgan's Law, the complement of the intersection of two sets is equal to the union of their complements: $(A \cap B)' = A' \cup B'$
   • Let's verify:
     • $A \cap B = \{3\}$
     • $A' = \{5, 6\}$ and $B' = \{1, 2, 4\}$
     • $A' \cup B' = \{1, 2, 4, 5, 6\}$
     • $(A \cap B)' = U - \{3\} = \{1, 2, 4, 5, 6\}$
     • This shows that $(A \cap B)' = A' \cup B'$.

5. (v) Find $A \Delta B$:

   • The symmetric difference $A \Delta B$ includes elements in $A \cup B$ but not in $A \cap B$. $A \Delta B = (A \cup B) - (A \cap B)$
   • $A \cup B = \{1, 2, 3, 4, 5, 6\}$
   • $A \cap B = \{3\}$
   • Therefore: $A \Delta B = \{1, 2, 4, 5, 6\}$

Would you like me to expand further or provide more details? Here are some related questions for deeper understanding:

1. What is the relation between union and intersection of sets?
2. How can we represent De Morgan’s laws using Venn diagrams?
3. Can you prove that $(A \Delta B)' = A' \cap B'$?
4. What are the applications of symmetric difference in probability?
5. How does the complement operation affect the cardinality of a set?

Tip: Always try using Venn diagrams to visualize set operations like union, intersection, and complements for a clearer understanding.