Set (Representation)

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Describing a Set

When we specify elements of a set, we are simply describing the set. The most common methods used to describe sets are given below.

  1. Semantic definition/ Intentional definition/ Verbal description

    It is also called intentional definition, using a rule or semantic description. This is called verbal description method. For Example

    1. A is the set of first four positive integers.
    2. B is the set of colors used in Nepali flag.
  2. Syntatic definition/ Extensional definition / Listing method

    This method is called roster notation or listing method done by listing each member of the set. This extensional definition is denoted by enclosing the list of members in curly brackets. For Example

    1. C = {4, 2, 1, 3}
    2. D = {blue, white, red}

    In this method, every element of a set must be unique; no two members may be identical. However, the order in which the elements of a set are listed is ignored (unlike for a sequence or tuple).
    Combining these two ideas into an example
    A={6, 11} , B= {11, 6}
    In the examples above, A = B.

  3. Venn-Diagram

    In this method, a diagram is used to represent the relationship among the sets, called Venn-diagram. It was named after an English philosopher John Venn (1834-1923). In a Venn-Diagram, rectangular region represent universal set, and other subsets usually by circular regions (sometimes, it can be expressed by ellipse, sphere etc). For Example
    If U={1,2,3,4,5,6,7,8,9,10}
    A={1,2,3,4,5}
    B={4,5,6,7,8}
    Then, the Venn-Diagram can be described as below.

  4. The set-builder notation
    1. A={x:x is the set of first four positive integers}
    2. B={x|x is the set of colors used in Nepali flag}

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

Exercise

तल दिइएको सङ्कलित संख्याका आधारमा प्रश्नहरूको उत्तर दिनुहोस्।
Answer the following questions based on the given collected numbers:
\(A=\{2, \pi, \frac{2}{7}, 0.11, 13, 45.\overline{33}, 19\}\)

  1. दिइएको सङ्कलन परिभाषित (well-defined) छ कि छैन, कारणसहित उल्लेख गर्नुहोस्।
  2. दिइएको सङ्कलनलाई सेट-बिल्डर (set-builder) रूपमा लेख्नुहोस्।

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