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 about sets, we need to use basic set operations and logical reasoning.
1. To prove the statement: "For all sets A, B, and C, if A - (B ∩ C) = ∅, then A - C ⊆ B."
We can use logical reasoning to prove this:
Assume that A - (B ∩ C) = ∅.
This means that there are no elements in A that are not present in (B ∩ C). Therefore, all elements of A are either in B or in C (or both).
Now, let's consider A - C.
If A - C is not empty, it means there exists at least one element in A that is not in C.
But since all elements of A are either in B or in C (or both), if there is an element in A that is not in C, it must be in B.
Therefore, A - C ⊆ B.
2. To prove the statement: "For all sets A, B, and C, if A ⊆ B ∪ C and B ⊆ C ∪ A, then A∆B ⊆ C."
We can use logical reasoning and set operations to prove this:
Assume that A ⊆ B ∪ C and B ⊆ C ∪ A.
We need to prove that A∆B ⊆ C.
Start by simplifying A∆B using the symmetric difference formula: A∆B = (A ∪ B) - (A ∩ B).
Let's consider an arbitrary element x in A∆B (x ∈ A∆B).
This means that x is in either A or B, but not both.
Case 1: x ∈ A
Since A ⊆ B ∪ C, it follows that x ∈ B ∪ C.
But x is not in B (as x ∈ A) and x is not in A ∩ B (by definition of symmetric difference).
Therefore, x ∈ C. Thus, x ∈ A∆B implies x ∈ C.
Case 2: x ∈ B
Similarly, since B ⊆ C ∪ A, it follows that x ∈ C ∪ A.
But x is not in A (as x ∈ B) and x is not in A ∩ B.
Therefore, x ∈ C. Thus, x ∈ A∆B implies x ∈ C.
In both cases, we have shown that if x ∈ A∆B, then x ∈ C. Hence, A∆B ⊆ C.
3. To prove the statement: "For all sets A, B, and C, if A∆B = A∆C, then B ⊆ C."
We can use logical reasoning and set operations to prove this:
Assume that A∆B = A∆C.
We need to prove that B ⊆ C.
Consider an arbitrary element x in B (x ∈ B).
We need to show that x ∈ C.
Since x is in B, it must be in either A or B - A (elements in B but not in A).
If x is in A, then it must be in A∆B and consequently in A∆C because A∆B = A∆C.
This means that x is in A∩C (by definition of symmetric difference).
Therefore, x ∈ C.
If x is in B - A, then it must be in A∆B and consequently in A∆C because A∆B = A∆C.
This means that x is in B - A∩C (by definition of symmetric difference).
Therefore, x ∈ C.
In both cases, we have shown that if x ∈ B, then x ∈ C. Hence, B ⊆ C.
By proving these statements using logical reasoning and basic set operations, we have established the relations between sets A, B, and C as mentioned in the given statements.