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
Thanks for your help.
To prove whether a statement like R or S is true or false for all sets A, B, and C, we need to use logical reasoning and set properties to reach a conclusion.
For statement R:
R states that if A is a subset of the union of B and C, and B is a subset of the union of C and A, then the symmetric difference of A and B is equal to C. To prove if R is true or false, we need to show that it holds for any possible choice of sets A, B, and C.
To begin proving R, we will start with the definition of a symmetric difference:
AΔB = (A - B) ∪ (B - A)
Assuming that R is true, we substitute AΔB with C:
C = (A - B) ∪ (B - A)
Now, let's consider the simplest case where A, B, and C are all empty sets. In this case, both A⊆B∪C and B⊆C∪A are true since the empty set is a subset of any set, and the union of empty sets is also an empty set.
Using these assumptions, we substitute A, B, and C with empty sets in the equation:
∅ = (∅ - ∅) ∪ (∅ - ∅)
∅ = ∅ ∪ ∅
∅ = ∅
This shows that the statement holds true for this particular case (i.e., when A, B, and C are all empty sets). However, to prove that R is true for all sets A, B, and C, we must show that it holds true for any other choice of sets as well. Here, we have only considered one particular case, so we cannot definitively conclude that R is true for all sets based on this alone.
To prove R for all sets, one would need to provide a formal proof using mathematical induction or logical deductions. It may also be helpful to counterexample to show that R is false for certain sets.
For statement S:
S states that if the symmetric difference of A and B is equal to the symmetric difference of A and C, then B is a subset of C. To prove if S is true or false, we need to show that it holds for any possible choice of sets A, B, and C.
Similarly to R, we can start by assuming S is true and substitute AΔB with AΔC:
AΔB = AΔC
Next, we consider the definition of the symmetric difference:
AΔB = (A - B) ∪ (B - A)
Substituting this definition into the equation, we have:
(A - B) ∪ (B - A) = (A - C) ∪ (C - A)
Now, we need to show that if AΔB = AΔC, B is a subset of C. To do this, one approach is to assume that B is not a subset of C and derive a contradiction. By assuming the negation of the statement, we can show that S is false.
For example, let's assume that B is not a subset of C. Then, there exists an element x such that x∈B and x∉C. Using this assumption, let's analyze the definitions of the symmetric difference:
(A - B) ∪ (B - A) = (A - C) ∪ (C - A)
Here, we can see that x∈(B - A) but x∉(C - A). Since x∈AΔB, this contradicts the assumption that AΔB is equal to AΔC.
As we have derived a contradiction, our initial assumption that B is not a subset of C must be false. Therefore, we can conclude that if AΔB = AΔC, then B is a subset of C.
Again, this proof is based on a particular assumption, and to prove S for all sets, a formal proof using mathematical induction or logical deductions is needed. Counterexamples can also be useful in showing that S is false for certain sets.