Answer to Question #313027 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.

E. How many students were taking Data Management and Discrete Mathematics but not Mathematics in the Modern World?

F. How many students did not take any of the mathematics subjects mentioned in the problem?

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 (E)

The number of students who were taking Data Management and Discrete Mathematics but not Mathematics in the Modern World were just 5 students.

Solution (F)

There were only two students who did not take any of the mathematics subjects mentioned in the problem.