Let A, B, and C be events. Find expressions for the following events: (a) Exactly one of the three events occurs. (b) Exactly two of the events occur. (c) One or more of the events occur. (d) Two or more of the events occur. (e) None of the events occur.

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) '