Answer:
a) (A∩B'∩C')U(A'∩B∩ C')U(A'∩B'∩C)
b) (A∩B∩C')U(A∩B'∩ C)U(A'∩B∩C)
c) A ∪ B ∪ C
d) (A∩B∩C')U(A∩B'∩C)U(A'∩B∩C)U(A∩B∩C)
e) (A ∪ B ∪ C)'
Explanation
Given the following events:
A, B, and C.
Unions, complement and intersection are defined as:
A ∪ B ∪ C= This represents event event where A, B, or C occurs. This is when at least one event occurs.
A ∩ B ∩ C = This represents event where A, B and C all occur
A' = This represents event where A does not occur.
a) Event where exactly one of the three events occurs: Here there will be 3 cases.
When A occurs, B and C do not occur; or
When B occurs, A and C do not occur; or
When C occurs, A and B do not occur;
(A∩B'∩C')U(A'∩B∩ C')U(A'∩B'∩C)
b) Event where exactly two of the events occur: Here,
When A and B occur, C does not occur; or
When A and C occur, B does not occur; or
When B and C occur, A does not occur
(A∩B∩C')U(A∩B'∩C)U(A'∩B∩C)
c) Event where one or more of the events occur: A ∪ B ∪ C (This means at least 1 event will occur)
d) Event where two or more of the events occur: (A∩B∩C')U(A∩B'∩C)U(A'∩B∩C)U(A∩B∩C) (This means at least 2 events will occur)
e) Event where None of the events occur: (A ∪ B ∪ C)'