Set
उदाहरणको लागी
\(A = \{a, e, i, o, u\}\) (1): The set of vowels
Set is a well defined collection of objects.
समुह भनेको राम्रोसँग परिभाषित गर्न सकिने वस्तुहरूको सङ्ग्रह हो।
How do we know if a set is well defined?
समूहमा "परिभाषित" भन्नाले सदस्यहरूलाई \(\in\) वा \(\notin\) प्रयोग गरि व्याख्या गर्न सकिने कुरालाई जनाउदछ। जस्तै,\(A = \{a, e, i, o, u\}\) (1): The set of vowels
Here, (1) मा दिइएको समूह \(A\) परिभाषित छ किनभने हामी भन्न सक्छौं की
\(a \in A , b \notin A\)
यँहा, \(\in\) को अर्थ "अन्तर्गत पर्छ" वा "सदस्य हो" भन्ने हुन्छ। यदि कुनै पनि कुरा दिएको समुहको सदस्य भएमा, हामी \(\in\) प्रयोग गर्छौं। र \(\notin\) को अर्थ "अन्तर्गत पर्दैन" वा "सदस्य होईन" भन्ने हुन्छ। यदि कुनै पनि कुरा दिएको समुहको सदस्य नभएमा, हामी \(\notin\) प्रयोग गर्छौं। जस्तै-
समुह \(A\) मा \(a\) पर्दछ, त्यसैले \(a \in A\)
समुह \(A\) मा \(b\) पर्दैन त्यसैले \(b \notin A\)
Member of set
समूहका सदस्यहरुलाई member भनिन्छ, जसलाई मझौला कोष्ठ \(\{\cdots\}\) भित्र राखिन्छ। समूहका सदस्यहरू भौतिक वस्तुहरू जस्तै किताब, कलम, व्यक्ति वा धारणात्मक वस्तुहरू जस्तै सङ्ख्या, विन्दु वा अन्य प्रकारका बस्तुहरु पनि हुन सक्छ। समूहका सदस्यहरू अंग्रेजी वर्णमालाका अक्षरहरू छन भने साना अक्षरहरू लेखेर जनाइन्छ ।
| Symbol | Name | Example | Explanation |
|---|---|---|---|
| \(\{ \}\) | Set | \(A = \{a,e,i,o,u\}\) | The set of vowels |
| \(\in\) | Membership | \(a \in A, e \in A, i \in A, o\in A,u \in A\) | The symbol \(\in\) denotes membership |
| \(\notin\) | Non-membership | \(5 \notin A, b \notin A\) | The symbol \(\notin\) denotes non-membership |
The marks of a few students of class 8 in a school are given below.
| Bidhi- 45 | Bidhan- 43 | Ram- 52 | Shyam- 49 | Pemba- 41 | Najir- 46 | Min- 51 | Najma- 48 |
- Can you make a set of 'talent students'? Give reason.
- Can you make a set of students with marks 'more than 45'? Give reason.
- Can you make a set of students with marks 'less than 43'? Give reason.
Let's say
\(A = \{Fe, Fo\}\) and \(B = \left\{1.3,\ \pi,\ \sqrt[3]{2},\ \frac{1}{3},\ 3.33\cdots\right\}\)
- Is the set \(A\) well defined? Give reason.
- Is the set \(B\) well defined? Give reason.
- Yes, the set \(A = \{Fe, Fo\}\) is well defined because the terms “Fe” and “Fo” are clearly specified.
- Yes, the set \(A = \{1.3,\ \pi,\ \sqrt[3]{2},\ \frac{1}{3},\ 3.33\cdots\}\) is well defined because the elements are clearly specified.
For a set to be well defined, its elements must be clearly identifiable and there should be no ambiguity to use membership element in the set.
\(Fe \in A\) and \(Fo \in A\)
\(1.3 \in B\) and \(4 \in B\) and so on.
तलको तालिकामा भएका समुहहरु "परिभाषित" छ वा छैन थाहा पाउनको लागी विचार गरि
\(True-T\) वा
\(False-F\) वा
\(\text{Not applicable}\)-NA
लेख्नुहोस र
समुहहरु ``परिभाषित" भएमा ``Yes" वा ``परिभाषित" नभएमा ``No" लेख्नुहोस।
Exercise
1. If \(A = \{2, 4, 6, 8, \cdots\}\), write 'true' or 'false'.
- \(6 \in A\)
- \(12 \in A\)
- \(5 \in A\)
- \(10 \notin A\)
- \(15 \notin A\)
- \(18 \notin A\)
- Is the set \(A\) well defined?
2. Use (✓) for well-defined collections, and (✗) for others.
- A collection of Nepali movies released in 2081 B.S.
- A collection of favourite Nepali movies released in 2081 B.S.
- A collection of smaller prime numbers less than 10.
- A collection of prime numbers less than 10.
3. Let's take a collection of any three "high mountains of Nepal" and answer:
- Is it a well-defined collection? Give reason.
- Is it a set? Give reason.
- Express it as a well-defined collection and list members.
4. How do you know if a set is well defined?
What are different ways to representation a Set?
समूहलाई साधारणतया चार प्रमुख तरिकाले प्रस्तुत गर्न सकिन्छ जस्तै (1) सूचीको रूपमा -roster form, (2) सङ्केतको रुपमा -set-builder form, (3) वर्णनात्मक रूपमा -descriptive form, र (4) भेन-चित्रको रुपमा -Venn-diagram form।| Method | Example | Explanation |
|---|---|---|
| Description | \(A\) is a set of multiples of 3 less than 15. | described by words. |
| Listing (or roster) | \(A = \{3, 6, 9, 12\}\) | elements are listed inside \(\{\}\). |
| Set-builder (or rule) | \(A = \{x : x \in \text{multiples of } 3, x < 15\}\) | variable \(x\) describe the properties |
-
roster form : \(A = \{a, e, i, o, u\}\)
The roster form (also called the "listing method"), explicitly lists all the elements of the set within curly braces \(\{\}\). Each element is written only once, and the order of elements does not matter.
For example, \(\{a,e,i,o,u\}\) is the same as \(\{u,o,i,e,a\}\). -
set builder form : \(A = \{x: x \text{ is a vowel}\}\)
The set builder form describes a set by specifying a property that all elements of the set must satisfy. It is written as \(\{x: \text{condition on } x\}\), which is read as "the set of all \(x\) such that the condition on \(x\) is true."
This method is useful when listing all elements is impractical or impossible. -
descriptive form : \(A = \{\text{set of vowels}\}\)
The descriptive form uses plain language to describe the set. Instead of listing elements or using mathematical notation, the set is described in words.
Here, the set \(A\) is described as "the set of vowels." This is a simple and intuitive way to convey the idea of the set without using formal mathematical notation. This method is often used for quick explanations or informal contexts. -
Venn diagram form
A Venn diagram is a visual representation of a set using circles or oval shapes, and the elements of the set can be shown inside the region. Venn diagrams are useful for visualizing relationships between sets, such as unions, intersections, and complements. They are particularly useful in problems involving multiple sets.
भेनचित्रको प्रयोग सन् 1880 मा गणितज्ञ John Venn ले गरेका थिए । उनकै नामबाट यस चित्रलाई भेन चित्र भनिएको हो ।
Exercise
तल दिइएको सङ्कलित संख्याका आधारमा प्रश्नहरूको उत्तर दिनुहोस्।
Answer the following questions based on the given collected numbers:
\(A=\{2, 3, 5, 7, 11, 13, 17, 19\}\)
- दिइएको सङ्कलन परिभाषित (well-defined) छ कि छैन, कारणसहित उल्लेख गर्नुहोस्।
- दिइएको सङ्कलनलाई समूह संकेत (set notation) मा लेख्नुहोस्।
- दिइएको सङ्कलनलाई सेट-बिल्डर (set-builder) रूपमा लेख्नुहोस्।
Write the given collection in set-builder form.
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