Question

in a survey of 374 coffee drinkers, it was found that 64 take only sugar, 82 take only cream, and 65 do not take sugar nor cream with their coffee. use a venn diagram and the principle of inclusion and exclusion to determine how many take: a. sugar and cream with their coffee? type your answer... b. cream? type your answer... c. neither sugar nor cream? type your answer... d. exactly one of these two additions? type your answer... e. at least one of these two additions? type your answer... 5 multiple choice 1 point which set - builder notation describes \\{ - 3,-2,-1,0,1,2\\}? \\( \\{ x | - 3 < x \leq 2 \\) where \\( x \\) is an integer \\} \\( \\{ x | - 3 \leq x \leq 2 \\) where \\( x \\) is an integer \\} \\( \\{ x | - 3 < x < 2 \\) where \\( x \\) is an integer \\} \\{ x | - 3 < x < 2 \\) where \\( x \\) is an integer \\}

Explanation:

Step1: Calculate the number of people who take at least one of sugar or cream

Total number of coffee drinkers \(n(T)=374\). Number of people who take neither \(n(\text{neither}) = 65\). So, \(n(\text{at least one})=n(T)-n(\text{neither})=374 - 65=309\)

Step2: Let \(n(S\cap C)=x\) (number of people who take both sugar and cream). Number of people who take only sugar \(n(\text{only }S)=64\), number of people who take only cream \(n(\text{only }C)=82\)

We know that \(n(\text{at least one})=n(\text{only }S)+n(\text{only }C)+n(S\cap C)\)
Substitute the values: \(309=64 + 82+x\)

Step3: Solve for \(x\)

\(x=309-(64 + 82)=309 - 146=163\) (a)

Step4: Calculate the number of people who take cream

\(n(C)=n(\text{only }C)+n(S\cap C)=82 + 163=245\) (b)

Step5: (c) is given as \(65\)

Step6: Calculate the number of people who take exactly one

\(n(\text{exactly one})=n(\text{only }S)+n(\text{only }C)=64+82 = 146\) (d)

Step7: (e) is \(309\)

For the set - builder notation:
The set \(\{-3,-2,-1,0,1,2\}\)
The inequality for set - builder notation: \(-3\leq x\leq2\) where \(x\) is an integer.

Answer:

a. \(163\)
b. \(245\)
c. \(65\)
d. \(146\)
e. \(309\)
For the multiple - choice: \(\{x|-3\leq x\leq2\text{ where }x\text{ is an integer}\}\) (the second option)