1. for any sets X and Y, we define the symmetric difference X ∆Y by:

X∆Y = (X - Y) ∪ (Y - X) = (X ∪ Y) - (X ∩ Y)
Prove the following:
-for all sets A,B and C, if A - (B ∩ C) = ∅ then A- C ⊆ B
- for all sets A,B and C, if A ⊆ B ∪ C and B ⊆ C ∪ A then A∆B ⊆ C
- for all sets A,B and C, if A∆B = A∆C then B ⊆C

To prove these statements, we will use the definition of the symmetric difference and basic set operations. Let's start with the first statement:

Statement 1: For all sets A, B, and C, if A - (B ∩ C) = ∅, then A - C ⊆ B.

Proof:
1. Assume A - (B ∩ C) = ∅.
2. By the definition of set difference, A - (B ∩ C) means all elements in A that are not in (B ∩ C).
3. If A - (B ∩ C) = ∅, this means that there are no elements in A that are not in (B ∩ C).
4. To show A - C ⊆ B, we need to prove that any element in A that is not in C must be in B.
5. Let x be an arbitrary element in A - C.
6. By definition, x is in A and not in C.
7. Since A - (B ∩ C) = ∅, it implies that any element in A must also be in (B ∩ C).
8. So, x must be in (B ∩ C).
9. Since x is not in C, it implies that x must be in B.
10. Therefore, A - C ⊆ B.

Next, let's move on to the second statement:

Statement 2: For all sets A, B, and C, if A ⊆ B ∪ C and B ⊆ C ∪ A, then A∆B ⊆ C.

Proof:
1. Assume A ⊆ B ∪ C and B ⊆ C ∪ A.
2. We want to show that any element in A∆B must also be in C.
3. By the definition of the symmetric difference, A∆B = (A - B) ∪ (B - A).
4. Let x be an arbitrary element in A∆B.
5. If x is in (A - B), then it is in A but not in B.
6. Since A ⊆ B ∪ C, it implies that x must be in B ∪ C.
7. If x is not in B, it must be in C.
8. Therefore, any element in (A - B) is also in C.
9. Now, consider the case where x is in (B - A), meaning it is in B but not in A.
10. Since B ⊆ C ∪ A, it implies that x must be in C.
11. Therefore, any element in (B - A) is also in C.
12. Hence, all elements in A∆B are in C, so A∆B ⊆ C.

Lastly, let's prove the third statement:

Statement 3: For all sets A, B, and C, if A∆B = A∆C, then B ⊆ C.

Proof:
1. Assume A∆B = A∆C.
2. To prove B ⊆ C, we need to show that any element in B must also be in C.
3. By the definition of the symmetric difference, A∆B = (A - B) ∪ (B - A) and A∆C = (A - C) ∪ (C - A).
4. Since A∆B = A∆C, it implies that (A - B) ∪ (B - A) = (A - C) ∪ (C - A).
5. By applying set operations, we can rewrite this equation as (A - B) ∪ (B - A) ∪ (C - A) = (A - C) ∪ (C - A) ∪ (B - A).
6. Simplifying both sides, we have (A - B) ∪ (C - A) = (A - C) ∪ (B - A).
7. Let x be an arbitrary element in B.
8. Since x is in B, it must be in either (A - B) or (C - A).
9. If x is in (A - B), it must also be in (A - C) due to their equality.
10. If x is in (C - A), it must also be in (B - A) due to their equality.
11. Hence, in both cases, x is in C.
12. Therefore, B ⊆ C.

By proving these three statements using logical reasoning and the definitions of set operations, we have demonstrated their validity.