Problem Solving (three sets)

Question

Eight blood types are shown by the eight regions of the Venn diagram shown here. Each circle represents one of three antigens: A, B or Rh. If A and B are both absent, the blood type is O. If Rh is present, the blood type is positive (+); otherwise, it is negative (-). The following table represents the blood types of 150 people.

Antigens	Number of people
A	60
B	27
Rh	123
A and B	12
A and Rh	46
B and Rh	17
A and B and Rh	9

1. A,B र Rh ले क्रमश: A, B र Rh एन्टीजेनलाई जनाउँछ भने सबै समुहलाई गणनात्मकता सङ्केतमा लेख्नुहोस् ।
   If A,B and Rh denote the set of people who have A, B र Rh Antigens in their blood respectively, write the cardinality notation of people for all groups.[1K]
2. मााथिको जानकारीलाई भेनचित्रमा प्रस्तुत गर्नुहोस् ।
   Show the above information in a Venn diagram. [1U]
3. कति जना मानिसहरुमा ओ नेगेटिभ रगत समुह रहेछन् ? गणना गर्नुहोस् ।
   How many people are having O negative blood? Calculate it. [3A]
4. रगत प्राप्त गर्दा, प्राप्तकर्तासँग दाताको सबै एन्टिजेनहरू हुनुपर्छ। अब, ओ पोजेटिभ भएको व्यक्तीले कति जना मानिसलाई रगत दान गर्न सक्दछ?
   When receiving a blood, the recipient must have all the antigens of the donor. If a person has O positive, to how many people he can donate the blood?[1HA]

Solution 👉 Click Here

Solution
Let
U is universal set of all antigens
Given that
A denote the set of people having A antigens
B denote the set of people having B antigens
Rh denote the set of people having Rh antigens
Thus
Cardinality of all surveyed people as Universal Set U is
\(n(U)=150\)
Now the itemwise solution are :

1. If A,B and Rh denote the set of people who have A, B र Rh Antigens in their blood respectively, write the cardinality notation of people for all blood groups.
   According the question,
   Cardinality of set, people having A, B and Rh antigens is
   \(n(A \cap B \cap Rh )=n(AB+)=9\)
   Cardinality of set, people having A and Rh antigens only is
   \(n_o(A \cap Rh )=n(A+)=46-9=37\)
   Cardinality of set, people having B and Rh antigens only is
   \(n_o(B \cap Rh )=n(B+)=17-9=8\)
   Cardinality of set, people having A and B antigens only is
   \(n_o(A \cap B )=n(AB-)=12-9=3\)
   Cardinality of set A, people having A antigens only is
   \(n_o(A)=n(A-)=60-37-9-3=11\)
   Cardinality of set B, people having B antigens only is
   \(n_o(B)=n(B-)=27-8-9-3=7\)
   Cardinality of set Rh, people having Rh antigens only is
   \(n_o(Rh)=n(O+)=123-37-9-8=69\)
   
2. Present the above information in a Venn-diagram.
   According the question,
   The Venn-diagram of given information are as follows.
3. How many people are having O negative blood? Calculate it.
   According to the Venn-diagram given above
   \(n(U)=150\)
   or\(69+11+7+37+3+8+9+x=150\)
   or\(x=6\)
   Therefore,
   The number of people having O negative blood is
   \(n \overline{(A \cup B \cup Rh)}=6\)
   
4. When receiving a blood, the recipient must have all the antigens of the donor. If a person has O+, to how many people he can donate the blood?
   According to the solution
   A person having O+ blood, can donate blood to
   people having Rh antigens
   people having O+,A+,B+, AB+ blood groups
   Which is
   \(n(Rh)=123\)
   Thus, A person having O+ blood, can donate blood to 123 people
   

This completes the solution