Final answer:
The statements relate to set theory in mathematics and are true, illustrating basic properties of union, intersection, and complements with respect to a universal set. Mutually exclusive sets have an intersection of zero, and this concept is important in probability calculations.
Step-by-step explanation:
The question deals with set operations and properties within the context of a universal set U. Let's examine each of the provided statements.
- (a) A∪(U\A)=U: This statement is true because the union of set A and the complement of A (everything in U not in A) gives us the entire universal set U.
- (b) A\(B∪C)=(A\B)∩(A\C): This is true due to the distributive property of sets. Subtracting B and C from A separately and then taking the intersection of the two results is equivalent to subtracting their union from A.
- (c) U\(U\A)=A: This statement is also true. Subtracting from U the set of elements that are in U but not in A leaves us with A itself.
It should be understood that mutually exclusive sets contain no common elements, and the probability of their intersection is zero. The set operations 'AND' (∩) and 'OR' (∪) correspond to intersection and union, respectively.
In the example given, A AND C are mutually exclusive, so their intersection would indeed be an empty set. When considering sets A and B, we don't assume they are mutually exclusive without evidence to support it, as the concept is crucial when calculating probabilities such as P(A OR B).