Question

a restaurant compiled the following information regarding 38 of its employees. of these employees, 11 cooked food, 13 washed dishes, 21 operated the cash register, 7 washed dishes and operated the cash register, 6 cooked food and washed dishes, 5 cooked food and operated the cash register, and 5 did all three jobs. complete parts a) through e) below.
a) how many of the employees only cooked food? (simplify your answer.)
b) how many of the employees only operated the cash register? (simplify your answer.)
c) how many of the employees washed dishes and operated the cash register but did not cook food? (simplify your answer.)
d) how many of the employees washed dishes or operated the cash register but did not cook food? (simplify your answer.)
e) how many of the employees did at least two of these jobs? (simplify your answer.)

Explanation:

Step1: Define sets

Let \(C\) be the set of employees who cooked food, \(D\) be the set of employees who washed dishes and \(R\) be the set of employees who operated the cash - register. We know \(n(C) = 11+6 + 5=22\), \(n(D)=13 + 7+5 = 25\), \(n(R)=21+7 + 5=33\), and \(n(C\cap D\cap R)=5\), \(n = 38\).

Step2: Solve part a

The number of employees who only cooked food is \(n(C)-n(C\cap D)-n(C\cap R)+n(C\cap D\cap R)\). The number of employees who cooked food and washed dishes is \(n(C\cap D)=6 + 5=11\), the number of employees who cooked food and operated the cash - register is \(n(C\cap R)=6 + 5 = 11\). So the number of employees who only cooked food is \(11\).

Step3: Solve part c

The number of employees who washed dishes and operated the cash - register but did not cook food is \(n(D\cap R)-n(C\cap D\cap R)=7\).

Step4: Solve part d

The number of employees who washed dishes or operated the cash - register but did not cook food:
First, \(n((D\cup R)\cap\overline{C})\). Using the principle of inclusion - exclusion, \(n(D\cup R)=n(D)+n(R)-n(D\cap R)=25 + 33-7 - 5=46\). The number of elements in \(D\cup R\) that are not in \(C\) is \(n((D\cup R)\cap\overline{C})=(21 + 7)+(13+7)-7=37\).

Step5: Solve part e

The number of employees who did at least two of these jobs:
\(n((C\cap D)\cup(C\cap R)\cup(D\cap R))\). Using the principle of inclusion - exclusion: \(n(C\cap D)=11\), \(n(C\cap R)=11\), \(n(D\cap R)=7 + 5=12\), \(n(C\cap D\cap R)=5\). Then \(n((C\cap D)\cup(C\cap R)\cup(D\cap R))=(11 + 11+12)-2\times5=24\).

Answer:

a. 11
c. 7
d. 37
e. 24