First, let's understand the symmetric difference operator. The symmetric difference of two sets P and Q, denoted as PΔQ, is defined as the set of elements that are in either P or Q, but not in both.

Now, let's analyze each statement using Venn diagrams.

I. P ∆ (Q ∪ R) = (P ∆ Q) ∪ (P ∆ R)
To prove this statement, we need to show that both sides of the equation contain the same elements.

- Let's start with the left-hand side (LHS) of the equation: P ∆ (Q ∪ R)
- This means we first take the union of sets Q and R, which includes all the elements that are in either Q or R.
- Next, we take the symmetric difference of set P with the union of sets Q and R, which includes elements that are in either P or (Q ∪ R), but not in both.
- This can be represented in a Venn diagram as follows:

```
__________
| |
| P |
|__________|

_________________
| |
| Q ∪ R |
|_______________|
```

- Now let's move on to the right-hand side (RHS) of the equation: (P ∆ Q) ∪ (P ∆ R)
- This means we first take the symmetric difference of sets P and Q, which includes elements that are in either P or Q, but not in both.
- Then, we take the symmetric difference of sets P and R, which includes elements that are in either P or R, but not in both.
- Finally, we take the union of these two symmetric differences, which includes elements that are in either (P ∆ Q) or (P ∆ R).
- This can be represented in a Venn diagram as follows:

```
__________
| |
| P |
|__________|

__________
| |
| Q |
|__________|

__________
| |
| R |
|__________|
```

By comparing the Venn diagrams, we can observe that the LHS and RHS do not represent the same sets. Therefore, statement I is FALSE.

II. P ∆ (Q ∪ R) = (P ∆ Q) ∩ (P ∆ R)
To prove this statement, we need to show that both sides of the equation contain the same elements.

- Let's start with the LHS of the equation: P ∆ (Q ∪ R)
- This means we first take the union of sets Q and R, which includes all the elements that are in either Q or R.
- Next, we take the symmetric difference of set P with the union of sets Q and R, which includes elements that are in either P or (Q ∪ R), but not in both.
- This can be represented in a Venn diagram as shown in the previous explanation.

- Now let's move on to the RHS of the equation: (P ∆ Q) ∩ (P ∆ R)
- This means we first take the symmetric difference of sets P and Q, which includes