Answer to Question #313005 in Discrete Mathematics for Jay

Question

Suppose that 53 of the 55 Information Technology students of University of Northern Philippines are taking atleast one of the mathematics subjects Mathematics in the Modern World, Discrete Mathematics, and Data management. Also suppose that: 24 taking Mathematics in the Modern World, 26 taking Discrete Mathematics, and 20 taking Data Management, 5 taking Mathematics in the Modern World and Discrete Mathematics, 7 taking Mathematics in the Modern World and Data Management, 8 taking Discrete Mathematics and Data Management.

A.   Fill in the correct number of students in each of the eight regions of the Venn Diagram.

B.   Find the number of students who are taking all three mathematics subjects.

C.   Determine the number of students who are not taking any of the three mathematics subjects?

D.   How many students were taking Mathematics in the Modern World as their only mathematics subject?

Answer 1

Solution

Let

Mathematics in Modern World is set A – Total student 24

Discrete Mathematics is set B – Total students 26

Data Management is set C – Total students 20

Since there are 53 students out of 55 who are taking at least one of the mathematics courses (taking one, two, or all three). It means there are two students who are not taking any of the mathematics courses.

The students taking both Mathematics in Modern World and Discrete Mathematics are 5

The students taking both Mathematics in Modern World and Data management is 7

The students taking both Discrete Mathematics and Data management are 8

We plot the Venn diagram as shown in the figure below 

Now according to the given conditions, we can write

x + w = 5                   …       (1)

y + w = 7                   …       (2)

z + w = 8                   …       (3)

24 + 26 – x – w + 20 – w – y – z = 53

17 = x + y +z + 2w   …       (4)

 

Adding (1) and (2)

x + y + 2w = 12        Replace in (4)

z + 12 = 17                

z = 5

Hence                        5 + w = 8 , w = 3,

                                    x + 3 = 5, x = 2,

                                    y + 3 = 7, y = 4.

Therefore, we update the Venn diagram accordingly as shown below.

Now

Solution (A)

The correct number of students in each of the eight regions of the Venn Diagram is shown above. Therefore, the number of students taking only

Mathematics in Modern World are 15, Discrete Mathematics is 16, and Data Management is 8.

Solution (B)

The number of students taking all the three subjects is 3.

Solution (C)

The number of students who are not taking any of the three mathematics subjects is 2.

Solution (D)

The students taking only the Modern World as their only mathematics subject are 15 students.