The problem defines a new set operation $A \Delta B$, which is known as the symmetric difference of two sets $A$ and $B$. The symmetric difference, denoted by $A \Delta B$, contains elements that are in either $A$ or $B$, but not in both. Mathematically:

$A \Delta B = \{x : (x \in A \, \text{and} \, x \notin B) \, \text{or} \, (x \in B \, \text{and} \, x \notin A)\}.$

Part (a):

You are asked to draw and shade Venn diagrams for the following three cases:

1. $A \Delta B$
2. $A \Delta A$
3. $A \Delta \emptyset$

After drawing the diagrams, you are asked to hypothesize about the outcomes of $A \Delta A$ and $A \Delta \emptyset$, and prove the results.

1. $A \Delta B$:

The symmetric difference $A \Delta B$ includes elements that belong to either $A$ or $B$, but not both. In a Venn diagram, this is represented by shading the parts of sets $A$ and $B$ that do not overlap (the regions that are exclusive to $A$ and $B$).

2. $A \Delta A$:

For the symmetric difference of a set with itself, $A \Delta A$, we look at the definition. If $x \in A$, then $x \notin A$ is false, so no elements can satisfy this condition. Therefore:

$A \Delta A = \emptyset.$

This means there are no elements that are in $A$ but not in $A$, so the result is the empty set. In a Venn diagram, no region would be shaded.

3. $A \Delta \emptyset$:

The symmetric difference between a set $A$ and the empty set $\emptyset$ consists of all elements that are in $A$ but not in $\emptyset$ (since no elements exist in $\emptyset$). Therefore:

$A \Delta \emptyset = A.$

In a Venn diagram, this corresponds to shading all of set $A$.

Hypothesis:

• $A \Delta A = \emptyset$: The symmetric difference of a set with itself is always the empty set.
• $A \Delta \emptyset = A$: The symmetric difference between a set and the empty set is the set itself.

Proof of the Results:

• Proof of $A \Delta A = \emptyset$:

  By definition, $A \Delta A = \{x \in A \text{ and } x \notin A \} \cup \{x \in A \text{ and } x \notin A \}$. Both conditions are contradictory, so no element can belong to $A \Delta A$. Hence, $A \Delta A = \emptyset$.

• Proof of $A \Delta \emptyset = A$:

  By definition, $A \Delta \emptyset = \{x \in A \text{ and } x \notin \emptyset\} \cup \{x \in \emptyset \text{ and } x \notin A\}$. Since no elements are in $\emptyset$, this simplifies to $A \Delta \emptyset = A$.

Do you have any questions, or would you like further clarification on any part of this?

Here are five follow-up questions you can explore:

1. How does the symmetric difference relate to the union and intersection of sets?
2. What happens when you take the symmetric difference of three sets, $A \Delta B \Delta C$?
3. Can the symmetric difference operation be generalized to more than two sets?
4. Is the symmetric difference operation commutative and associative? Can you prove it?
5. What are some practical applications of the symmetric difference in real-world problems?

Tip: Symmetric difference is useful in areas such as data analysis where you need to identify differences between sets, such as comparing changes between versions of datasets.