Math  /  Data & Statistics

QuestionA survey of 135 freshmen busjness students at a local university produced the results listed below. 32 took English: 37 took math; 36 took religion; 16 took English but not math; 11 took math and religion: 12 took English and religion; 6 took all three Using the same reasoning as in the previous problem, we can find that the Purple region has a cardinality of 6.
Question: Find the Cardinality of the Brown region using: "32 took English; and "16 took English but not math;" Hint: English but not Math is Yellow + Purple
Hine DO NOT try to use a formula. Think your way through it. And DRAW the Venn Diagram!

Studdy Solution

STEP 1

1. We are given a survey of 135 freshmen business students with various combinations of subjects taken.
2. We have to find the cardinality of the Brown region using the information about students taking English and those taking English but not math.
3. The Brown region likely represents students who took English but did not take math or religion.

STEP 2

1. Identify and represent the given data on a Venn diagram.
2. Use the given information to determine the number of students in the relevant regions of the Venn diagram.
3. Calculate the cardinality of the Brown region based on the Venn diagram and the provided data.

STEP 3

Draw a Venn diagram with three intersecting circles representing students taking English (E), Math (M), and Religion (R).

STEP 4

Label the regions of the Venn diagram according to the given information: - EE: English - MM: Math - RR: Religion - EME \cap M: Students taking both English and Math - ERE \cap R: Students taking both English and Religion - MRM \cap R: Students taking both Math and Religion - EMRE \cap M \cap R: Students taking all three subjects

STEP 5

Place the number of students in the appropriate regions of the Venn diagram based on the given data: - 6 students in the region EMRE \cap M \cap R (Purple region, taking all three subjects) - 16 students in the region EME \setminus M (Yellow + Purple, taking English but not Math)

STEP 6

Determine the number of students taking only English (Brown region), which is represented as E(MR)E \setminus (M \cup R).

STEP 7

Use the information that 32 students took English to find the number of students taking only English: E=32 |E| = 32

STEP 8

Subtract the number of students who took English but not Math from the total number of students who took English, to find those who took English and Math. EM=E(EM) |E \cap M| = |E| - (|E \setminus M|) EM=3216=16 |E \cap M| = 32 - 16 = 16

STEP 9

Subtract the number of students who took all three subjects (Purple) from the number taking English and Math to find those who took only English and Math (Blue region). EMR=EMEMR |E \cap M \setminus R| = |E \cap M| - |E \cap M \cap R| EMR=166=10 |E \cap M \setminus R| = 16 - 6 = 10

STEP 10

Subtract the number of students who took all three subjects (Purple) from the number taking English and Religion to find those who took only English and Religion (Green region). ERM=EREMR |E \cap R \setminus M| = |E \cap R| - |E \cap M \cap R| ERM=126=6 |E \cap R \setminus M| = 12 - 6 = 6

STEP 11

Use the information that 16 students took English but not Math (Yellow + Purple region) to find the number of students who took English and Religion but not Math (Yellow region). ERM=6 |E \cap R \setminus M| = 6

STEP 12

Finally, calculate the number of students who took only English (Brown region) by subtracting those who took English and Math (Blue), English and Religion (Green), and all three subjects (Purple) from the total number of students who took English. E(MR)=E(EMR+ERM+EMR) |E \setminus (M \cup R)| = |E| - (|E \cap M \setminus R| + |E \cap R \setminus M| + |E \cap M \cap R|) E(MR)=32(10+6+6) |E \setminus (M \cup R)| = 32 - (10 + 6 + 6) E(MR)=3222=10 |E \setminus (M \cup R)| = 32 - 22 = 10
Solution: The cardinality of the Brown region is 10.

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