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 if statement R is true or false for all sets A, B, and C, we need to show that the statement holds true for all possible combinations of sets A, B, and C.
To help understand the statement R, let's break it down:
R: For all sets A, B, and C, 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 R, we need to show that the symmetric difference of A and B (denoted AΔB) is always equal to C.
The symmetric difference of two sets A and B is defined as the set of elements that are in either A or B, but not in both A and B. In mathematical notation, AΔB = (A∪B) - (A∩B).
To prove R, we can use the following steps:
Step 1: Assume A, B, and C are arbitrary but fixed sets.
Step 2: Assume A is a subset of B∪C and B is a subset of C∪A.
Step 3: Show that AΔB = C.
To show that AΔB = C, we need to prove two sub-claims:
Sub-claim 1: AΔB is a subset of C.
Sub-claim 2: C is a subset of AΔB.
Sub-claim 1: To prove AΔB is a subset of C, we need to show that every element in AΔB is also in C.
Let x be an arbitrary element in AΔB. This means that x is in either A or B, but not in both.
Case 1: x is in A
Since A is a subset of B∪C, x is also in B∪C. If x were in B, it would violate the definition of AΔB. Therefore, x must be in C.
Case 2: x is in B
Since B is a subset of A∪C, x is also in A∪C. If x were in A, it would violate the definition of AΔB. Therefore, x must be in C.
In both cases, we have shown that x is in C for every element x in AΔB. Therefore, AΔB is a subset of C.
Sub-claim 2: To prove C is a subset of AΔB, we need to show that every element in C is also in AΔB.
Let y be an arbitrary element in C. We need to show that y is in AΔB, meaning y is in either A or B, but not in both.
Since A is a subset of B∪C, and y is in C, y is also in B∪C. If y were in B, it would violate the definition of AΔB. Therefore, y must be in A.
Thus, we have shown that every element y in C is also in AΔB. Therefore, C is a subset of AΔB.
Since both sub-claims hold true, we can conclude that AΔB = C.
Therefore, 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 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 the statement holds true for all possible combinations of sets A, B, and C.
We can prove S by assuming A, B, and C are arbitrary but fixed sets and show that if AΔB = AΔC, then B is a subset of C.
To prove S, we can use the following steps:
Step 1: Assume A, B, and C are arbitrary but fixed sets.
Step 2: Assume AΔB = AΔC.
Step 3: Show that B is a subset of C.
To show that B is a subset of C, we need to prove that every element in B is also in C.
Let x be an arbitrary element in B. We need to show that x is in C.
Since AΔB = AΔC, every element in B must be in A but not in C, or vice versa.
If x were in A but not in C, then x would be in AΔB but not in AΔC, which would contradict the assumption that AΔB = AΔC.
Therefore, x cannot be in A but not in C. This means that x must be in C.
Since x is an arbitrary element in B and we have shown that it is also in C, we can conclude that B is a subset of C.
Therefore, statement S is true for all sets A, B, and C.
Hope this helps! Let me know if you have any further questions.