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 will use set algebra and logical reasoning.
1. For all sets A, B, and C, if A - (B ∩ C) = ∅, then A - C ⊆ B.
Proof:
To prove the statement, we need to show that if A - (B ∩ C) = ∅, then every element in A - C is also in B.
Let's assume that A - (B ∩ C) = ∅.
This means that there is no element in A that is not in (B ∩ C), therefore, every element in A is either in B or in C.
Now, we want to show that A - C ⊆ B.
Suppose there is an element x in A - C, which means that x is in A, but not in C.
Since every element in A is either in B or in C, x must be in B.
Therefore, we have shown that if A - (B ∩ C) = ∅ then A - C ⊆ B.
2. For all sets A, B, and C, if A ⊆ B ∪ C and B ⊆ C ∪ A, then A∆B ⊆ C.
Proof:
To prove the statement, we need to show that if A ⊆ B ∪ C and B ⊆ C ∪ A, then every element in A∆B is also in C.
Let's assume that A ⊆ B ∪ C and B ⊆ C ∪ A.
We know that A∆B = (A - B) ∪ (B - A).
If an element x is in A∆B, then it must be either in A - B or in B - A.
Case 1: x ∈ A - B
This means that x is in A but not in B. Since A ⊆ B ∪ C, every element in A is either in B or in C. But x is not in B, so it must be in C.
Case 2: x ∈ B - A
This means that x is in B but not in A. Similarly, since B ⊆ C ∪ A, every element in B is either in C or in A. But x is not in A, so it must be in C.
In both cases, we have shown that every element in A∆B is also in C.
Therefore, we have proved that if A ⊆ B ∪ C and B ⊆ C ∪ A, then A∆B ⊆ C.
3. For all sets A, B, and C, if A∆B = A∆C, then B ⊆ C.
Proof:
To prove the statement, we need to show that if A∆B = A∆C, then every element in B is also in C.
Let's assume that A∆B = A∆C.
This means that all elements that are in either A but not in B or in B but not in A are also in either A but not in C or in C but not in A.
To prove that B ⊆ C, we need to show that any element x in B is also in C.
Suppose there is an element x in B but not in C.
This means that x is in B but not in A and not in C.
Since x is not in A∆B, x must be in A∆C. This implies that x is either in A∆C but not in A or in A∆C but not in C.
However, this is a contradiction because we already assumed that all elements in A∆B are also in A∆C.
Therefore, there cannot exist an element x in B but not in C, so B ⊆ C.
Hence, we have proved that if A∆B = A∆C, then B ⊆ C.