150 adults in Center City were randomly surveyed regarding breakfast habits. Respondents were asked what fruit(s) they ate for breakfast. Let U= the universe (the set containing all respondents) A= set of all adults who responded they eat an apple with breakfast, and B= set of all adults who responded they eat blueberries with breakfast. Survey results are reported in the Venn diagram above.

Brandon Monroe

Brandon Monroe

Answered question

2022-08-07

150 adults in Center City were randomly surveyed regarding breakfast habits. Respondents were asked what fruit(s) they ate for breakfast. Let U= the universe (the set containing allrespondents) A= set of all adults who responded they eat an apple with breakfast, and B= set of all adults who responded they eat blueberries with breakfast. Survey results are reported in the Venn diagram above.
a. How many adults responded that they eat an apple with breakfast?
(This value = n ( A ). "the number of elements in set A")
b. How many of the adults responded that they eat blueberries with breakfast?
(This value = n ( B ). "the number of elements in set B")
c. How many of the adults responded that they eat both apples and blueberries with breakfest?
(This value = n ( A B ), "the number of elements in the intersection" of sets A and B")
d. How many of the adults responded that they eat apples or blueberries with breakfest?
(This value = n ( A B ), "the number of elements in the union of sets A and B")
e. How many adults responded that they eat apples but not blueberries with breakfast?
(This value = n ( A B ), "the number of elements in the intersection of set A and the complement of set B")
f. How many adults responded that they eat blueberries but not apples with breakfast?
(This value = n ( A B ), "the number of elements in the intersection of the complement of set A and set B").
g. How many adults responded that they eat either apples or blueberries with breakfest, but not both?
(This value = [ n ( A B ) n ( A B ) ], "the number of elements in the union of sets A and B minus the number of elements in the intersection of sets A and B")
h. How many adults responded that they eat neither apples nor blueberries with breakfast?
(This value = n ( A B ), "the number of elements in the intersection of the complement of set A and the complement of set B")

Answer & Explanation

Isaias Archer

Isaias Archer

Beginner2022-08-08Added 11 answers

Step 1
As per the Venn diagram, the universal set U consists of 150 adults who took part in the survey.
a. The set A consists of n ( A ) = 40 + 47 = 87 adults who responded that they eat an apple with the breakfast. The answer is 87.
b. The set B consists of n ( B ) = 32 + 47 = 79 adults who responded that they eat blueberries with breakfast. The answer is 79.
Step 2
c. The set A B consists of n ( A B ) = 47 adults who responded that they eat both an apple and blueberries with breakfast. The answer is 47.
d. The set A B consists of n ( A B ) = 40 + 47 + 32 = 119 adults who responded that they eat either an apple or blueberries with breakfast. The answer is 119.
Dorsheele0p

Dorsheele0p

Beginner2022-08-09Added 5 answers

Step 1
e. n ( A B ) = 40 adults responded that they eat apples, but not blueberries with breakfast. The answer is 40.
f. n ( A B ) = 32 adults responded that they eat blueberries, but not apples with breakfast. The answer is 32.
Step 2
g. 40 + 32 = 72 adults responded that they eat either apples or, blueberries with breakfast, but not both. The answer is 72.
h. n ( A B ) = n [ ( U A ) ( U B ) ] = n ( A B ) = 150 ( 40 + 47 + 32 ) = 150 119 = 31 adults responded that they eat neither apples, nor blueberries with breakfast. The answer is 31.

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