# elementary sets theory [closed]

A survey was conducted on the newspaper readership of three dailies: The Mirror, The Citizen and The Times.

The following data was obtained: the number of the people who read The Mirror and The Times were 19, The Citizen and The Mirror were 15, The Citizen and The Times were 14. Those who read all the three were found to be 4 people only.

Determine the number of people who read The Mirror only, The Citizen or The Times but not the Mirror, and total number of people interviewed if 5 read none of the newspaper.

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## closed as not a real question by Asaf Karagila, Thomas, rschwieb, Arkamis, NorbertOct 22 '12 at 19:45

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I must have accidentally clicked the answers.yahoo bookmark. Oh wait... –  The Chaz 2.0 Oct 22 '12 at 17:25
help solve the problem. –  KENNETH MUTEMBEI MBAABU Oct 23 '12 at 7:04

The data are insufficient.

Since $4$ people read all three papers, there were $19-4=15$ who read the Mirror and the Times but not the Citizen, $15-4=11$ who read the Citizen and the Mirror but not the Times, and $14-4=10$ who read the Citizen and the Times but not the Mirror. This makes a total of $15+11+10=36$ people who read exactly two of the papers, so there are $36+4=40$ people who read at least two of the papers. We’re also told that there are $5$ people who read none of the papers.

Let $n$ be the number of people surveyed, $c$ the number who read only the Citizen, $m$ the number who read only the Mirror, and $t$ the number who read only the Times. Then we know that $$n=c+m+t+45\;,\tag{1}$$ but that’s all that we can say without further information: fill in any non-negative integers for $c,m$, and $t$, compute $n$ by $(1)$, and the result will be consistent with the information in the problem.

$$\begin{array}{c} \text{Newspapers read}&\text{Number reading}\\ \hline \text{CMT}&4\\ \text{CM}&11\\ \text{CT}&10\\ \text{MT}&15\\ \text{C}&c\\ \text{M}&m\\ \text{T}&t\\ \text{none}&5\\ \end{array}$$

In particular, $m$ read only the Mirror, $c+t+10$ read the Citizen or the Times (or both) but not the Mirror, and $c+m+t+45$ were interviewed.

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thankyou. But how do you get c,m and t? –  KENNETH MUTEMBEI MBAABU Oct 23 '12 at 7:01
@KENNETH: As I said, there isn’t enough information in the problem to determine them. You can let them be any non-negative integers that you like, and the results will be consistent with the data given. –  Brian M. Scott Oct 23 '12 at 7:06
:Thankyou the post was helpful to me. –  KENNETH MUTEMBEI MBAABU Oct 24 '12 at 5:54

This is a case where you should apply the Inclusion–exclusion principle, where the sets are the readers of the respective journals.

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substituting what in the principle formula? –  KENNETH MUTEMBEI MBAABU Oct 23 '12 at 7:03
@KENNETH: Venn diagram with three sets $C$, $M$, $T$. $|M\cap T|=19$, etcetera. The answer by Brian shows you do not have all data to get a clear answer to the total number of people involved. –  Hendrik Jan Oct 23 '12 at 8:14