mathbalarka said:
"Proofing the equation" refers to the process of using logical reasoning and mathematical operations to demonstrate that the equation A=B Union C & B ∩ C=Ø implies the statement (A\B=C) is true.
A=B Union C means that the set A contains all elements that are in either B or C (or both). In other words, the elements of A are the combination of the elements in B and C.
B ∩ C=Ø means that the sets B and C have no common elements. In other words, the intersection of B and C is an empty set.
The equation A=B Union C & B ∩ C=Ø implies that every element in A is also in either B or C (or both), but not exclusively in B. This means that the difference between A and B (represented by A\B) is equal to the set C, as every element in C is not in B but is in A.
One example could be the equation A={1,2,3,4} & B={2,4} & C={3,5}. We can show that (A=B Union C & B ∩ C=Ø) implies (A\B=C) by first proving that A=B Union C and B ∩ C=Ø are true. This can be done by listing out the elements of A, B, and C and demonstrating that they satisfy the definitions of a union and an empty intersection. Then, we can use the same elements to show that (A\B=C) is also true, as the difference between A and B is the set {1,3}, which is equal to C.