To solve the expression $(A \Delta B) \Delta (A \Delta C)$, where $\Delta$ represents the symmetric difference between two sets, let’s recall the definition of symmetric difference:

$A \Delta B = (A \cup B) \setminus (A \cap B)$

In words, the symmetric difference between two sets $A$ and $B$ is the set of elements that are in either $A$ or $B$, but not in both.

Step-by-Step Breakdown:

We are tasked with finding $(A \Delta B) \Delta (A \Delta C)$.

1. Calculate $A \Delta B$:

$A \Delta B = (A \cup B) \setminus (A \cap B)$ This is the set of elements that are in $A$ or $B$, but not both.

2. Calculate $A \Delta C$:

$A \Delta C = (A \cup C) \setminus (A \cap C)$ This is the set of elements that are in $A$ or $C$, but not both.

3. Now compute $(A \Delta B) \Delta (A \Delta C)$:

We take the symmetric difference of the two results: $(A \Delta B) \Delta (A \Delta C) = ((A \Delta B) \cup (A \Delta C)) \setminus ((A \Delta B) \cap (A \Delta C))$ This set contains the elements that are in either $A \Delta B$ or $A \Delta C$, but not in both.

Conclusion:

The expression simplifies to the symmetric difference between the symmetric differences of $A$ with $B$ and $A$ with $C$. In general, the outcome will depend on the specific sets $A$, $B$, and $C$, but the general structure is:

$(A \Delta B) \Delta (A \Delta C)$

Would you like further details or a specific example to clarify this?

5 Related Questions:

1. How do the union and intersection operations affect symmetric difference outcomes?
2. What are the properties of symmetric differences that can simplify complex expressions?
3. How does the symmetric difference of two sets relate to their complements?
4. What happens if sets $B$ and $C$ are equal in the expression $(A \Delta B) \Delta (A \Delta C)$?
5. Can this expression be extended to more than three sets?

Tip:

Symmetric difference can be visualized using Venn diagrams to see which elements are in the distinct regions of the sets. This helps understand how symmetric differences combine!