Set (समूह)
$\underline{\large \textbf{1. Important formulae:}}\\ 1. \normalsize \space n(A \cap B)=n(A)+n(B)-n(A \cup B)\\ 2. \normalsize \space n(U)=n(A)+n(B)-n(A \cap B)+ n \overline{(A \cup B)}\\ 3. \normalsize \space n_o(A)=n(A)-n(A \cap B)\\ 4. \normalsize \space n_o(B)=n(B)-n(A \cap B)\\ 5.\normalsize \space n(U)=n_o(A)+n_o(B)+n(A \cap B)+ n \overline{(A \cup B)}\\ 6. \normalsize \space n(A \cup B)=n_o(A)+n_o(B)+n(A \cap B)\\ 7. \normalsize \space n(\overline{A \cup B)}=n(U)-n(A \cup B)\\ $
Shaded the following parts of the sets by clicking the set notation.
Including three sets.$ 8. \normalsize \space n(U)=n(A)+n(B)+n(C)-n(A \cap B)-n(B \cap C)-n(C \cap A) + n(A \cap B \cap C) + n \overline{(A \cup B \cup C)}\\ 9. \normalsize \space n(A \cup B \cup C)=n(A)+n(B)+n(C)-n(A \cap B)-n(B \cap C)-n(C \cap A) + n(A \cap B \cap C)\\ $ $\underline{\large \textbf{2. Illustrations:}}\\ $
1.
Here
मानाै, A र B ले क्रमशः स्वदेशी चिया र बिदेशी चिया मन पराउने मानिसहरुकाे समूहलाई जनाउँछ । U ले सर्वव्यापक समुह जनाउँछ भने
$n(U)=600\\
n(A)=300\\
n(B)=250\\
n(A \cap B)=150\\
\overline{n(A \cup B)}=?\\
\text{By the formula,}\\
\space n(U)=n(A)+n(B)-n(A \cap B) + \overline{n(A \cup B)}\\
\space or, 600=300+250-150+ \overline{n(A \cup B)}\\
\space or, 600=550-150 + \overline{n(A \cup B)}\\
\space or, 600=400 + \overline{n(A \cup B)}\\
\space or, \overline{n(A \cup B)}=600-400\\
\space \therefore \overline{n(A \cup B)}=200. ans.\\
$
माथिकाे तथ्याङ्कलाई भेन चित्रमा देखाउँदाः
Next Problem.
2.
Here
मानाै, A र B ले क्रमशः रेडियाे सुन्न र टेलिभिजन हेर्न मन पराउने मानिसहरुकाे समूहलाई जनाउँछ । U ले सर्वव्यापक समुह जनाउँछ भनेः
(Let A and B be the two non-empty sets who like to listen the radio and to watch the television respectively. If U Denotes universal set, then):
$n(U)=100 \% \\
n(A)=55 \% \\
n(B)=65 \% \\
n(A \cap B)=35 \% \\
\overline{n(A \cup B)}=?\\
$
(i) माथीकाे जानकारीलाई भेनचित्रमा देखाउँदा (Presenting in a Venn-Diagram):
(i) रेडियाे सुन्न र टेलिभिजन हेर्न मन नपराउने प्रतिशत (percentage of the people who do not like to listen the radio as well as to watch the television) $\overline{n(A \cup B)}$:
From the Venn-Diagram (भेन चित्रबाट):
$ \space \therefore \overline{n(A \cup B)}=15 \% $ ans.
Next Problem.
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