Answer to Question #99967 in Discrete Mathematics for John

Question

In a class, 20 students study agric, 21 study biology and those who study chemistry are 5 more than those who study agric. it is also known that 3 students study agric only and 4 students study only chemistry. Half as many students who study chemistry only study biology only and twice as many students who study exactly one of the subjects study 2 of the 3 subjects.
If one third of the students who study biology study all there subjects and the number of students who study agric and chemistry only is equal to the number of students who do not study any of the 3 subjects.

a) Illustrate the above information on a vane diagram
b) How many students
i) are in the class
ii) study at least 2 subjects
iii) study at most one subject
iv) Do not study any of the subjects

Answer 1

a)

Vann (Venn) diagramm - is a diagram that shows all possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, and sets as regions inside closed curves. A Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. The points inside a curve labelled S represent elements of the set S, while points outside the boundary represent elements not in the set S. This lends to easily read visualizations; for example, the set of all elements that are members of both sets S and T, S ∩ T, is represented visually by the area of overlap of the regions S and T. In Venn diagrams the curves are overlapped in every possible way, showing all possible relations between the sets.

2 students study only biology, because "half as many students who study chemistry only study biology only",half of 4 is 2.

"one third of the students who study biology study all there subjects", 21 * (1/3) = 7 students who study biology study all there subjects

"study chemistry are 5 more than those who study agric" : 20+5=25 study chemistry.

"twice as many students who study exactly one of the subjects study 2 of the 3 subjects"

3 + 4 + 2 =9, 9*2=18

"who study exactly one of the subjects study 2 of the 3 subjects" - chemistry and biology (z) or chemistry and agrik (y) or agrik and biology (x)

Let x,y,z - number.

(II) x+y+z=18=9*2=(3 + 4 + 2)*2

3+x+y+7=20

x+y=10

y+z+7+4=25

y+z=14

x+z+7+2=21

x+z=12

=> because x+y+z=18, (x+y)+z=18

z=8

x+(y+z)=18

x=4

y+(x+z)=18

y=6

"number of students who study agric and chemistry only is equal to the (IV)number of students who do not study any of the 3 subjects." 

(IV)"agric and chemistry only " is "y" = 6

b) Class=3+2+4+4+6+8+6+7=40

(III) is "(IV) + only most one subject"

(III) 6+3+2+4=15

(II) study at least 2 subjects" is all (three) or two subject

all: 7

two: x+y+z=18

18+7=25