Set (Cardinality)

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Set

Cardinality of set

मानौ \(A\) एउटा समुह हो, समुह \( A \) मा भएका सबै सदस्यहरुको सँख्यालाई cardinality (or cardinal number ) of \( A \) भनिन्छ ।
यसलाई \(n(A) \) अथवा \( |A|\) अथवा \( \text{Card}(A)\) संकेतले जनाईन्छ।
जस्तै
If \( A = \{x: x < 4, x \in \mathbb{W} \} \) then \( A =\{0, 1, 2, 3\}\) and \(n (A) = 4\).
सामान्य रूपमा, समुहको cardinality ले तलका गुणहरू जनाउदछ।
  1. प्रत्येक समूहको एक मात्र cardinal सङ्ख्या हुन्छ।
  2. दुई वटा equivalent समूहको cardinal सङ्ख्या एउटै हुन्छ।
  3. गन्ती गर्न सकिने समूहको cardinal सङ्ख्या \(n\) हुन्छ।
  4. गणनायोग्य (denumerable) समूहको cardinal सङ्ख्या ‘एलेफ नल’ \(\aleph_0\) हुन्छ।
  5. गन्ती गर्न नसकिने (uncountable) समूहको cardinal सङ्ख्या \(c\) हुन्छ।

Find the cardinality of following set.

  1. \(A = \{1, 4, 9, 16\}\)

  2. \(B = \{1, 2, 3, 4, 5, \ldots, 15\}\)

  3. \(C = \{2, 4, 6, 8\}\)

  4. \(D = \{0, 1, 2, 3, 4, 5, \ldots, 10\}\)

  5. \(E = \{1, 2, 4, 5, 10, 20\}\)

A

What is the cardinal number of set A?

कक्षाकोठामा भएका बिद्यार्थी वा बस्तु लाई प्रयोग गरि बनाईएका फरक फरक समुहहरुको cardinality सँख्या पत्ता लगाउन लगाउनुहोस।

Let \(P = \{x : x \text{ is a natural number, } x \leq 20\}\), \(Q = \{x : x \text{ is a factor of } 20\}\) and \(R = \{z : z \text{ is a multiple of } 3, z ≤ 20\}\), then.
  1. List the elements of the sets \(P,Q,R\)
  2. List the common elements of the sets \(P,Q\)
  3. Identify the cardinality of common elements of the sets \(P,Q\)
  1. The elements of the sets \(P, Q, R\) are
    \(P = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}\)
    \(Q = \{1, 2, 4, 5, 10, 20\}\)
    \(R = \{3, 6, 9, 12, 15, 18\}\)
  2. The common elements of the sets \(P\) and \(Q\) is
    \( \{1, 2, 4, 5, 10, 20\} \)
  3. The cardinality of common elements of the sets \(P, Q\) is
    \(6\)
The factor-cards given below.
1 2 3 4 6 12
1 2 3 6 9 18
  1. Write the set of factors of 12 (\(F_{12}\)) in listing method.
  2. Write the set of factors of 18 (\(F_{18}\)) in listing method.
  3. Write the set of factors of 18 (\(F_{18}\)) in set-builder method.
  4. List the common elements of \(F_{12}\) and \(F_{18}\) in a separate set \(P\) and write its cardinal number.
  1. The set of factors of 12 (\(F_{12}\)) in listing method is
    \(F_{12} = \{1, 2, 3, 4, 6, 12\}\)
  2. The set of factors of 18 (\(F_{18}\)) in listing method is
    \(F_{18} = \{1, 2, 3, 6, 9, 18\}\)
  3. The set of factors of 18 (\(F_{18}\)) in set-builder method is
    \(F_{18} = \{x : x \text{ is a positive integer and } x \text{ divides } 18\}\)
    or\(F_{18} = \{x : x \mid 18,\ x \in \mathbb{N}\}\)
  4. The common elements of \(F_{12}\) and \(F_{18}\) in a separate set \(P\) is
    \(P = \{1, 2, 3, 6\}\)
    The cardinal number of \(P\) is
    \(n(P) = 4\)
The multiple-cards given below show first five multiples of 4 and 7.
4 8 12 4 16 20
7 14 21 28 35
  1. Write the set of the first five multiples of 4 (\(M_4\)) and 7 (\(M_7\)) in listing method.
  2. Write the sets \(M_4\) and \(M_7\) in set-builder method.
  3. Write a set \(A\) taking the common elements of \(M_4\) and \(M_7\).
  4. What type of set is the set \(A\)? Describe.
  1. The set of the first five multiples of 4 (\(M_4\)) and 7 (\(M_7\)) in listing method are
    \(M_4 = \{4, 8, 12, 16, 20\}\)
    \(M_7 = \{7, 14, 21, 28, 35\}\)
  2. The sets \(M_4\) and \(M_7\) in set-builder method are
    \(M_4 = \{x : x = 4n,\ n \in \mathbb{N},\ 1 \leq n \leq 5\}\)
    \(M_7 = \{x : x = 7n,\ n \in \mathbb{N},\ 1 \leq n \leq 5\}\)
  3. The set \(A\) taking the common elements of \(M_4\) and \(M_7\) is
    \(A = \{\}\) or \(A = \varnothing\)
  4. The type of set \(A\) is an empty set (also called null or void set) because it contains no elements.

Types of sets

समुहका प्रकारहरु निम्नानुसार छन।
  1. Empty set
    कुनै पनि सदस्य (तत्व) नभएको समूहलाई रिक्त (empty) वा शून्य (null) वा रित्तो (void) समूह भनिन्छ, र यसलाई \(\{ \}\) वा \(\phi\) ले जनाइन्छ।जस्तै
    \(A = \{ x: x \text{ is a natural number less than } 1\} = \phi \)
  2. Singleton set
    एउटा मात्र सदस्य (तत्व) भएको समूहलाई एकल समूह भनिन्छ।जस्तै
    \(A = \{ x: x \text{ is even prime}\} = \{2\} \)
  3. Finite set
    गन्ती गर्न सकिने सीमित संख्याहरु भएको समूहमा लाई finite set भनिन्छ।जस्तै
    \(A = \{ \text{ whole numbers less than } 10\} = \{ 0,1,2,3,4,5,6,7,8,9\} \)
  4. Infinite set
    गन्ती गर्न नसकिने असीमित संख्याहरु भएको समूहमा लाई infinite set भनिन्छ।जस्तै
    \(B = \{ x: x \in \mathbb{W}, x = 2n\} = \{ 0, 2, 4, 6, \ldots\} \)

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