Question

a survey of 105 college students was taken to determine where they got the news about whats going on in the world. of those surveyed, 47 got the news from newspapers, 52 from television, and 9 from both newspapers and television. construct a venn diagram and determine the cardinality for each region. use the completed venn diagram to answer the following questions. how many got the news from only newspapers? n(newspapers only)=38 how many got the news from only television? n(television only)=43 how many got the news from newspapers or television? n(newspapers or television)=90 how many did not get the news from either newspapers or television? n(neither)=□

Explanation:

Step1: Recall the principle of set - theory

Let \(N\) be the set of students who get news from newspapers and \(T\) be the set of students who get news from television. We know that \(n(N\cup T)=n(N)+n(T)-n(N\cap T)\) and the total number of students \(n = 105\). We are given \(n(N) = 47\), \(n(T)=52\) and \(n(N\cap T)=9\).

Step2: Calculate \(n(N\text{ only})\)

The number of students who get news only from newspapers \(n(N\text{ only})=n(N)-n(N\cap T)\). Substituting the values, we have \(n(N\text{ only})=47 - 9=38\).

Step3: Calculate \(n(T\text{ only})\)

The number of students who get news only from television \(n(T\text{ only})=n(T)-n(N\cap T)\). Substituting the values, we get \(n(T\text{ only})=52 - 9 = 43\).

Step4: Calculate \(n(N\cup T)\)

Using the formula \(n(N\cup T)=n(N\text{ only})+n(T\text{ only})+n(N\cap T)\), we have \(n(N\cup T)=38 + 43+9=90\).

Step5: Calculate \(n(\text{Neither})\)

The number of students who get news from neither newspapers nor television \(n(\text{Neither})=105 - n(N\cup T)\). Substituting \(n(N\cup T) = 90\), we get \(n(\text{Neither})=105 - 90=15\).

Answer:

15