Specification Grid Grade 910
S.N.  Areas  Working hours  Knowledge  Understanding  Application  Higher ability  Total items  questions  Marks  
Items  Marks  Items  Marks  Items  Marks  Items  Marks  
1.  Sets  12  1  1  1  1  1  3  1  1  4  1  6 
कक्षागत सिकाइ उपलब्धि ९ र १०
क्र.स.  विषयवस्तुको क्षेत्र  कक्षा ९  कक्षा १० 
१.  समूह  समूहका क्रियाहरु गर्न र भेन चित्रमा प्रस्तुत गर्न समूहको गणनात्मकता पत्ता लगाउन  समूहका क्रियाहरू, भेनचित्र र गणनात्मकताको प्रयोग गरी तीनओटासम्म समूहसँग सम्बन्धित व्यावहारिक समस्याहरू समाधान गर्न 
Scope and Sequence of Contents of Grade 9
समूह समूहका क्रियाहरू संयोजन, प्रतिच्छेदन, फरक र पुरक (तीनओटासम्म समूह)
 समूहको गणनात्मकता
Parts of Three Sets
मानौ, सर्वव्यापक समुह U को उपसमुहहरु A,B र C छन भने तिन वटा समुहहरु समावेस भएका समस्याहरु समाधान गर्न तलको भेन चित्र प्रयोग गर्नुहोस। (Let A, B and C are the subsets of an universal set U, then use the following Venndiagram to solve problems related to three sets.
 \( n_o(A)=p\)
\(n(ABC)=p\)
This part is also known as A difference with B and C as denoted by ABC. This parts represents the cardinality (or elements) which lies in only in A but niether in B nor in C.  \( n_o(B)=q\)
\( n(BCA)=q\)
This part is also known as B difference with C and A as denoted by BCA. This parts represents the cardinality (or elements) which lies in only in B but niether in C nor in A.  \( n_o(C)=r\)
\(n(CAB)=r\)
This part is also known as C difference with A and B as denoted by CAB. This parts represents the cardinality (or elements) which lies in only in C but niether in B nor in A.  \( n_o(A \cap B)=s\)
\(n((A \cap B)C)=s\)
This part is also known as intersection of A and B, only. This parts represents the cardinality (or elements) which lies in only intersection of A and B but NOt in C.  \( n_o(B \cap C)=t\)
\( n((B \cap C)A)=t\)
This part is also known as intersection of B and C, only. This parts represents the cardinality (or elements) which lies in only intersection of B and C but NOT in A.  \( n_o(A \cap C)=u\)
\( n(A \cap C)B)=u\)
This part is also known as intersection of A and C, only. This parts represents the cardinality (or elements) which lies in only intersection of A and C but NOT in B.  \(n(A \cap B \cap C)=v\)
This part is also known as intersection of A , B and C. This parts represents the cardinality (or elements) which lies in A, B and C, in all three sets.  \(\overline{AUBUC}=w\)
This part is also known as complement of union of A , B and C. It is also denoted by \( (A \cup B \cup C)'\) or \( (A \cup B \cup C)^c\). This parts represents the cardinality (or elements) which does NOT lier on either A or B or C.
Cardinality of Set
The concept and notation of Cardinality are due to Georg Cantor who defined the notion of cardinality and realized that sets can have different cardinalities. In summary,
 The cardinality of finite set A is \(n(A)\)
 The cardinality of countable set is \(\aleph_0\) (read as alephnaught or alephzero or alephnull)
 The cardinality of uncountable set is \(𝑐\) (read as continuum)
The cardinality of a set A is the number of elements of the set A . The cardinality of a set A is usually denoted by n(A) but it can also be denoted as Card(A). For example:
 If \( A = \{x: x< 4, x \in W \}\) then A = {0, 1, 2, 3} and n (A) = 4
 If B = { letters in the word “mathematics”} then B = {m, a, t, h, e, i, c, s} and n(B) = 8.
Arithmetic of Cardinality
Arithmetic of cardinality in sets refers to the mathematical operations that involve counting the number of elements (cardinality) within sets. When solving verbal problems involving sets, you might encounter situations where you need to perform arithmetic operations such as addition, subtraction, multiplication, and division on the cardinalities of sets to find the desired information.
Here are some common scenarios where arithmetic of cardinality comes into play when solving verbal problems related to sets:

Union of Sets: When we need to find the total number of elements in the union of two or more sets, we use the concept of cardinality. For example, if we have sets A and B, the cardinality of their union (A ∪ B) can be calculated by adding the cardinalities of A and B and then subtracting the cardinality of their intersection (A ∩ B) to avoid double counting any shared elements.
n(A ∪ B) = n(A) + n(B)  n(A ∩ B) 
Complements: The complement of a set A with respect to a larger set U (universal set) contains all elements in U that are not in A. You can calculate the cardinality of the complement by subtracting the cardinality of A from the cardinality of U.
n(A') = n(U)  n(A)  Subtraction of Sets: When you want to find the number of elements in one set that are not in another set, you can use subtraction of cardinalities. For instance, if you have sets A and B, the cardinality of the difference A  B is found by subtracting the cardinality of B from the cardinality of A.
n(A  B) = n(A)  n(B)
Test your Understandings
From the Venndiagram given below, find the cardinal number given sets.
 \(n(\phi) \)
 \(n(AB) \)
 \(n(BA) \)
 \(n(A \cap B) \)
 \(n(A \cup B)' \)
 \(n(A) \)
 \(n(B) \)
 \(n(A') \)
 \(n(B)' \)
 \(n(A \triangle B) \)
 \(n(A \triangle B)' \)
 \(n(AUB) \)
 \(n(A_0)' \)
 \(n(B_0)' \)
 \(n(A \cap B)' \)
 \(n(U) \)
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