Let R be the statement : for all sets A, B and C, if A⊆B∪C and B ⊆C∪A then AΔB = C.
Q: Is R true for all sets A, B and C? Prove answer
Let S be the statement: for all sets A, B, and C, if AΔB = AΔC then B⊆C.
Q: Is S true? prove answer
To prove whether statement R is true for all sets A, B, and C, we will have to use logical reasoning and the definitions of set operations.
Let's analyze statement R step by step:
R: For all sets A, B, and C, if A⊆B∪C and B ⊆C∪A, then AΔB = C.
To prove whether R is true, we need to show that whenever the conditions A⊆B∪C and B ⊆C∪A hold, the conclusion AΔB = C is also true.
Proof:
1. Assume that A⊆B∪C and B ⊆C∪A are true.
2. We want to show that AΔB = C.
3. Definition of AΔB: AΔB = (A∖B) ∪ (B∖A)
4. Since A⊆B∪C, every element in A must either be in B or in C. Therefore, (A∖B) = ∅, as there are no elements in A that are not in B.
5. Similarly, since B⊆C∪A, every element in B must either be in C or in A. Therefore, (B∖A) = ∅, as there are no elements in B that are not in A.
6. Based on the previous observations, AΔB = ∅∪∅ = ∅.
7. Therefore, for A⊆B∪C and B ⊆C∪A, AΔB = C is true.
Since we were able to establish that AΔB = C holds whenever A⊆B∪C and B ⊆C∪A are true, we can conclude that statement R is true for all sets A, B, and C.
Now let's move on to statement S:
S: For all sets A, B, and C, if AΔB = AΔC, then B⊆C.
To prove whether S is true, we need to show that whenever the condition AΔB = AΔC holds, the conclusion B⊆C is also true.
Proof:
1. Assume that AΔB = AΔC is true.
2. To show B⊆C, we will assume an arbitrary x∈B and prove that x∈C.
3. Since x∈B, we have two cases to consider:
a. If x∉A, then x must be in B∖A, as x∉A implies x∈B∖A. Since AΔB = AΔC, this means x must also be in C∖A. Therefore, x∈C.
b. If x∈A, then x must be in A∖B, as x∈A implies x∉B. Since AΔB = AΔC, this means x must also be in A∖C. Therefore, x∈C (since x∉B implies x∈C).
4. Based on both cases, we have shown that for any x∈B, x∈C. Therefore, B⊆C.
Since we were able to establish that B⊆C holds whenever AΔB = AΔC, we can conclude that statement S is true for all sets A, B, and C.