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Solving Problems Involving Sets

In this section we look at some examples involving sets.


Example: Write the elements of the following sets:

To write down the elements of S, note that W represents the set of positive integers and zero. Therefore, we get:

Now, we write the elements of F:

In order to write down the elements of Qwe use the definition for d and evaluate the elements:

The set K, is defined in terms of the set of natural numbers NThe set of natural numbers is the set of positive integers greater than and equal to one. Thus, we have the following elements in K:


Example: Consider the sets given in the previous example, find the following sets:

We get the following sets:


Example: Suppose ,,

a) First, we find


b) We get:


Example: In a class of 100 students, 35 like chemistry and 45 like physics and 10 like both. How many like either of them and how many like neither? Show these values on a Venn diagram.


Let's summarize the information given in the question:
Total number of students is n(T)=100
Number of students that like physics is n(P)=45
Number of students that like chemistry is n(C)=35
Therefore, total number of students that like both is:

Number of students that like neither is:


The Venn digram for this question is given by:



Example: There are 30 students in a class. Among them, 8 students are learning both French and Spanish. A total of 18 students are learning French. If every student is learning at least one language, how many students are learning Spanish in total?
We summarize the information as follows:
Total number of students is n(T)=30
Number of students that learn both French and Spanish is n(F and S)=8
Number of students that only learn French is n(F)=18-8=10
Number of students that are learning Spanish is 30-10=20
Number of students that only learn Spanish is 20-8=12


The Venn diagram for this problem is:



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