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Magazine Chapter 2: Q8E (page 57)
An engineering construction firm is currently working on power plants at three different sites. Let Aidenote the event that the plant at site i is completed by the contract date. Use the operations of union, intersection, and complementation to describe each of the following events in terms of \({A_1}\), \({A_2}\), and \({A_3}\), draw a Venn diagram, and shade the region corresponding to each one.
a. At least one plant is completed by the contract date.
b. All plants are completed by the contract date.
c. Only the plant at site 1 is completed by the contract date.
d. Exactly one plant is completed by the contract date.
e. Either the plant at site 1 or both of the other two plants are completed by the contract date.
Short Answer
a. The Venn diagram is represented as,
b. The Venn diagram is represented as,
c. The Venn diagram is represented as,
d. The Venn diagram is represented as,
e. The Venn diagram is represented as,
Step by step solution
Given information
Let \({A_i}\) represents the event that the plant at site i is completed by the contract date.
Describe the events by using the operations
a.
Let U represent the event that at least one plant is completed by the contract date.
The operation that will be used to describe the event U is union.
Mathematically,
\(U = {A_1} \cup {A_2} \cup {A_3}\)
Construction a Venn diagram
The Venn diagram representing the event U is given as,
Describe the events by using the operations for part (b).
b.
Let M be the event that all plants are completed by the contract date.
The operation that will be used to describe the event M is intersection.
Mathematically,
\(M = {A_1} \cap {A_2} \cap {A_3}\)
Construction of a Venn diagram
The Venn diagram representing the event M is given as,
Describe the events by using the operations for part (c)
c.
Let N be the event that only the plant at site 1 is completed by the contract date.
In the provided event only the area of\({A_1}\)will be shaded.
The operation that will be used to describe the event N is intersection and complementation.
Mathematically,
\(N = {A_1} \cap A_2^C \cap A_3^C\)
Construction of a Venn diagram
The Venn diagram representing the event N is given as,
Describe the events by using the operations for part (d)
d.
Let R be the event that exactly one plant is completed by the contract date.
The operation that will be used to describe the event R is intersection, complementation and union.
Mathematically,
\(R = \left( {{A_1} \cap A_2^C \cap A_3^C} \right) \cup \left( {A_1^C \cap {A_2} \cap A_3^C} \right) \cup \left( {A_1^C \cap A_2^C \cap {A_3}} \right)\)
Construction of a Venn diagram
The Venn diagram representing the event R is given as,
Describe the events by using the operations for part (e)
e.
Let T represent the event that either the plant at site 1 or both of the other two plants are completed by the contract date.
The operation that will be used to describe the event T is intersection, complementation and union.
Mathematically,
\(T = {A_1} \cup \left( {{A_2} \cap {A_3}} \right)\)
Construct a Venn diagram
The Venn diagram representing the event T is given as,
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