[নোটঃ এই আর্টিকেলটি (5) Statistics and D.I. বিভাগের অধীনে 〈3〉Overlapping and Venn Diagrams চ্যাপ্টারের অন্তর্গত, যা 〈5.3.a〉চিহ্ন দিয়ে প্রকাশ করা হয়েছে]

## 〈5.3.a〉ডাবল-সেটম্যাট্রিক্স এবং ভেন ডায়াগ্রাম

##### Overlapping Sets

সাধারণতঃ ভেনচিত্র বা ভেনডায়াগ্রাম ব্যবহার করে এ ধরণের অঙ্ক সমাধান করা হয়। এ ছাড়া ভেনডায়াগ্রামের পরিবর্তে ডাবল-সেটম্যাট্রিক্স নিয়ে কিভাবে ওভারল্যাপিং সেট এর সমাধান করতে হয় আমরা এখানে সে বিষয়টা দেখাবো। মনে রাখবেন, ডাবল-সেটম্যাট্রি· ব্যবহৃত হয় শুধুমাত্র দুইটি গ্রুপ এর জন্যে। গ্রুপ সংখ্যা তিন হলে ভেনডায়াগ্রাম লাগবে।

Of \(\)30\(\) integers, \(\)15\(\) are in set \(\)A\(\), \(\)22\(\) are in set \(\)B\(\), and \(\)8\(\) are in both set \(\)A\(\) and \(\)B\(\). How many of the integers are in NEITHER set \(\)A\(\) nor set \(\)B\(\)?

উপরের টেবিল অনুসারে আমরা মানগুলো বসিয়ে ফেললামঃ

এবার খালি ঘরগুলোতে মান বসানো হলোঃ

অনুরূপ আরেকটি সমস্যা দেখা যাকঃ

\(\)70\(\)% of the guests at Company \(\)X\(\)‘s annual holiday party are employees of Company \(\)X\(\). \(\)10\(\)% of the guests are women who are not employees of Company \(\)X\(\). If half the guests at the party are men, what percent of the guests are female employees of Company \(\)X\(\)?

First, fill in \(\)100\(\) for the total number of guests at the party. Then, fill in the other information given in the problem: \(\)70\(\)% of the guests are employees, and \(\)10\(\)% are women who are not employees. We also know that half the guests are men. (Therefore, we also know that half the guests are women.)

Next, use subtraction to fill in the rest of the information in the matrix:

\(\)100-70=30\(\) guests who are not employees

\(\)30-10=20\(\) men who are not employees

\(\)50-10=40\(\) female employees

\(\)40\(\)% of the guests at the party are female employees of Company \(\)X\(\). Note that the problem does not require us to complete the matrix with the number of male employees, since we have already answered the question asked in the problem. However, completing the matrix is an excellent way to check your computation. The last box you fill in must work both vertically and horizontally(যেমন এখানে \(\)100\(\) সংখ্যাটি \(\)70+30\(\) এভাবে এবং \(\)50+50\(\) এভাবে গঠিত হতে হবে).

** \(\)3\(\)-Set Problems: Venn Diagrams**

Problems that involve \(\)3\(\) overlapping sets can be solved by using a Venn Diagram. The three overlapping sets are usually \(\)3\(\) teams or clubs, and each person is either on or not on any given team or club. That is, there are only \(\)2\(\) choices for any dub: *member* or *not*.

Workers are grouped by their areas of expertise and are placed on at least one team. 20 workers are on the Marketing team, \(\)30\(\) are on the Sales team, and \(\)40\(\) are on the Vision team. \(\)5\(\) workers are on both the Marketing and Sales teams, \(\)6\(\) workers are on both the Sales and Vision teams, \(\)9\(\) workers are on both the Marketing and Vision teams, and \(\)4\(\) workers are on all three teams. How many workers are there in total?

In order to solve this problem, use a Venn Diagram.

A Venn Diagram should be used ONLY for problems that involve three sets. Stick to the double-set matrix for two-set problems.

Begin your Venn Diagram by drawing three overlapping circles and labeling each one.

Notice that there are \(\)7\(\) different sections in a Venn Diagram. There is one innermost section(\(\)A\(\)) where all \(\)3\(\) circles overlap. This contains individuals who are on all \(\)3\(\) teams.

There are three sections (\(\)B, C\(\), and \(\)D\(\)) where \(\)2\(\) circles overlap. These contain individuals who are on \(\)2\(\) teams. There are three non-overlapping sections (\(\)E, F\(\), and \(\)G\(\)) that contain individuals who are on only \(\)1\(\) team.

Venn Diagrams are easy to work with, if you remember one simple rule: **Work from the Inside Out.**

That is, it is easiest to begin by filling in a number in the innermost section (\(\)A\(\)). Then, fill in numbers in the middle sections (\(\)B, C\(\), and \(\)D\(\)). Fill in the outermost sections (\(\)E, F\(\), and \(\)G\(\)) last.

**First:** Workers on all \(\)3\(\) teams: Fill in the inner most circle. This is given in the problem as \(\)4\(\).

**Second:** Workers on \(\)2\(\) teams: Here we must remember to subtract those workers who are on all \(\)3\(\) teams. For example, the problem says that there are \(\)5 \(\)workers on the Marketing and Sales teams. However, this includes the \(\)4\(\) workers who are on all three teams.Therefore, in order to determine the number Marketingof workers who are on the Marketing andSales teams exclusively, we must subtract the 4 workers who are on all three teams. We are left with \(\)5-4=1\(\).

The number of workers on the Marketing and Vision teams exclusively is \(\)9-4=5\(\). The number of workers on the Sales and Vision teams exclusively is \(\)6-4=2\(\).

** ****Third: **Workers on \(\)1\(\) team only: Here we must remember to subtract those workers who are on \(\)2\(\) teams and those workers who are on \(\)3\(\) teams. For example, the problem says that there are \(\)20\(\) workers on the Marketing team. But this includes the \(\)1\(\) worker who is on the Marketing and Sales teams, the \(\)5\(\) workers who are on the Marketing and Vision teams, and the \(\)4\(\) workers who are on all three teams. We must subtract all of these workers to find that there are \(\)20- 1- 5-4=10\(\) people who are on the Marketing team exclusively. There are \(\)30-1-2-4=23\(\) people on the Sales team exclusively. There are \(\)40-2-5-4=29\(\) people on the Vision team exclusively.

In order to determine the total, just add all \(\)7\(\) numbers together = \(\)74\(\) total workers.