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 the given statements, we need to use the properties of set operations and logical deduction. Let's prove each statement one by one.

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

To prove this statement, we need to show that if A - (B ∩ C) is empty, then every element in A - C is also in B.

Proof:
1. Assume A - (B ∩ C) is empty, which means there are no elements in A that are not in B ∩ C.
2. Assume x is an arbitrary element in A - C.
3. By definition, x is in A and not in C.
4. Let's consider the two cases:
a. If x is also in B, then x is in B ∩ C, which contradicts the assumption A - (B ∩ C) is empty.
b. If x is not in B, then x is in A and not in B, which implies that x is in A - (B ∩ C).
5. Since both cases led to contradictions, our initial assumption that A - (B ∩ C) is empty must be false.
6. Therefore, A - C ⊆ B.

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

To prove this statement, we need to show that if A and B are subsets of B ∪ C and C ∪ A respectively, then every element in the symmetric difference A∆B is also in C.

Proof:
1. Assume x is an arbitrary element in A∆B.
2. By definition of symmetric difference, x is either in (A - B) or (B - A).
3. Let's consider the two cases:
a. If x is in (A - B), then x is in A but not in B.
- Since A ⊆ B ∪ C, x is either in B or in C.
- If x is in B, then it contradicts the assumption x is in (A - B).
- Therefore, x must be in C.
b. If x is in (B - A), then x is in B but not in A.
- Since B ⊆ C ∪ A, x is either in C or in A.
- If x is in A, then it contradicts the assumption x is in (B - A).
- Therefore, x must be in C.
4. In both cases, x is in C.
5. Thus, we have shown that for every element x in A∆B, x is in C, which implies A∆B ⊆ C.

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

To prove this statement, we need to show that if the symmetric difference of A and B is equal to the symmetric difference of A and C, then every element in B is also in C.

Proof:
1. Assume x is an arbitrary element in B.
2. If x is in A∆B, then x is in either (A - B) or (B - A).
3. Since A∆B = A∆C, x must also be in A∆C.
4. By definition of symmetric difference, x is in either (A - C) or (C - A).
5. We can infer that if x is in (A - B) or (B - A), x must also be in (A - C) or (C - A), respectively.
6. If x is in (A - C), then x is not in C.
7. If x is in (C - A), then x is in C.
8. Since x cannot be simultaneously in (A - C) and (C - A), we conclude that x must be in C.
9. Therefore, every element x in B is also in C, which means B ⊆ C.

By proving these statements, we have established the validity of the given claims.