Set (Representation)

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

1. 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 \(\{u,o,i,e,a\}\).
2. 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.
3. 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.
4. 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 ।

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Exercise

तल दिइएको सङ्कलित संख्याका आधारमा प्रश्नहरूको उत्तर दिनुहोस्।
Answer the following questions based on the given collected numbers:
\(A=\{2, 3, 5, 7, 11, 13, 17, 19\}\)

1. दिइएको सङ्कलन परिभाषित (well-defined) छ कि छैन, कारणसहित उल्लेख गर्नुहोस्।
2. दिइएको सङ्कलनलाई समूह संकेत (set notation) मा लेख्नुहोस्।
3. दिइएको सङ्कलनलाई सेट-बिल्डर (set-builder) रूपमा लेख्नुहोस्।
   Write the given collection in set-builder form.