Set (Subset)

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Subset

The universal set, denoted by \( U \), is the set that contains all the elements under consideration in a given context. Every other set is a subset of this universal set.

Let the universal set be
\(U = \{ \text{apple}, \text{banana}, \text{mango}, \text{orange}, \text{grape} \}\)
Let set \( A \) be the set of fruits given as
\(A = \{ \text{apple}, \text{mango}, \text{grape} \}\)
Then \(U\) is universal set, and \( A\) is subset.



Subset

कुनै समुह \( A \) को प्रत्येक सदस्य अर्को समुह \( B \) को पनि सदस्य हो भने, समुह \( A \) लाई समुह \( B \) को उपसमुह भनिन्छ।
यसलाई \( A \subset B \) भनेर लेखिन्छ र “ \( B \) को उपसमुह \( A \) ” भनेर पढिन्छ।
जस्तै: यदि \( A = \{1, 2, 3\}, B = \{3, 4, 5,6\} \) र \( C = \{1, 2, 3, 4, 5\} \) छन् भने
\( A \subset C \) तर \( B \not\subset C \).(चित्र 1 र 2 हेर्नुहोस्)


In the system of real numbers, with usual notation, the relation between sets are given as
\(\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset\mathbb{Q} \subset \mathbb{R}\).
Define each of the set \(\mathbb{N} , \mathbb{W}, \mathbb{Z} ,\mathbb{Q} , \mathbb{R}\) with one example on each.

  1. \(\mathbb{N}\)
    Example: \(5\)
  2. \(\mathbb{W}\)
    Example: \(0\)
  3. \(\mathbb{Z}\)
    Example: \(-3\)
  4. \(\mathbb{Q}\)
    Example: \(\frac{2}{3}\)
  5. \(\mathbb{R}\)
    Example: \(\sqrt{2}\)
  6. The Venn-diagram of \(\mathbb{N} , \mathbb{W}, \mathbb{Z} ,\mathbb{Q} , \mathbb{R}\) are given below.


Proper and Improper Subset

In general, there are two types of subsets, they are proper subset and improper subset.
  1. उपयुक्त उपसमुह (Proper subset)
    यदि \( A \subset B \) र \( A \ne B \) भने \( A \) लाई \( B \) को उपयुक्त उपसमुह भनिन्छ। यस अवस्थामा, \( B \) लाई \( A \) को सुपर समुह भनिन्छ।
    a set B which contains elements from A but, not all, is called proper subset of A
    If \(B=\{1, 5, 8, 9, 10\}\) , then proper subset of B are
    Number of Elements Subsets Count
    0 ∅ 1
    1 {1}, {5}, {8}, {9}, {10} 5
    2 {1, 5}, {1, 8}, {1, 9}, {1, 10}, {5, 8}, {5, 9}, {5, 10}, {8, 9}, {8, 10}, {9, 10} 10
    3 {1, 5, 8}, {1, 5, 9}, {1, 5, 10}, {1, 8, 9}, {1, 8, 10}, {1, 9, 10}, {5, 8, 9}, {5, 8, 10}, {5, 9, 10}, {8, 9, 10} 10
    4 {1, 5, 8, 9}, {1, 5, 8, 10}, {1, 5, 9, 10}, {1, 8, 9, 10}, {5, 8, 9, 10} 5
    Total Subsets 32
  2. अनुपयुक्त उपसमुह (Improper subset)
    समुहको परिभाषाबाट परम्परागत रूपमा नै, शून्य समुह र समुह आफैंलाई अनुपयुक्त उपसमुह पनि भनिन्छ। त्यसैले, यदि \( A \subset B \) र \( A = B \) भएमा \( A \) लाई \( B \) को अनुपयुक्त उपसमुह भनिन्छ। यसलाई \( A \subseteq B \) ले जनाईन्छ।
    a set B which contains all elements of A, is called improper subset of A
    If\(B=\{1, 5, 8, 9, 10\}\), then improper subset of B is
    \(B=\{1, 5, 8, 9, 10\}\)
NOTE
  1. शून्य समुह \( \phi \) प्रत्येक समुहको उपसमुह हो।
  2. प्रत्येक समुहको (खाली समुह बाहेक) कम्तीमा दुईवटा उपसमुहहरू हुन्छन्।
  3. \( n \) वटा सदस्य भएको समुहको सम्भावित उपसमुहहरु \( 2^n \) वटा हुन्छ, जसको समुहलाई Power Set भनिन्छ।


If \(A=\{a,e,i,o,u\}\), then

  1. find all subsets consisting no element
    \(\{\}\) or \(\phi\)
  2. find all subsets consisting 1 element
    \(\{a\}, \{e\}, \{i\}, \{o\}, \{u\}\)
  3. find all subsets consisting 2 elements
    \(\{a,e\}, \{a,i\}, \{a,o\}, \{a,u\}, \{e,i\}, \{e,o\}, \{e,u\}, \{i,o\}, \{i,u\}, \{o,u\}\)
  4. find all subsets consisting 3 elements
    10 subsets, e.g., \(\{a,e,i\}, \{a,e,o\}, \dots\))
  5. find all subsets consisting 4 elements
    5 subsets, e.g., \(\{a,e,i,o\}, \{a,e,i,u\}, \dots\))


If \(A=\{a\}, B=\{a,b\}, C=\{a,b,c\}, D=\{a,b,c,d\}\), then

  1. find all subsets of \(A\)
    Number of Elements Subsets Count
    0 ∅ 1
    1 {a} 1
    Total Subsets 2
  2. find all subsets of \(B\)
    Number of Elements Subsets Count
    0 ∅ 1
    1 {a}, {b} 2
    2 {a, b} 1
    Total Subsets 4
  3. find all subsets of \(C\)
    Number of Elements Subsets Count
    0 ∅ 1
    1 {a}, {b}, {c} 3
    2 {a, b}, {a, c}, {b, c} 3
    3 {a, b, c} 1
    Total Subsets 8
  4. find all subsets of \(D\)
    16 subsets
    Number of Elements Subsets Count
    0 ∅ 1
    1 {a}, {b}, {c}, {d} 4
    2 {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d} 6
    3 {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d} 4
    4 {a, b, c, d} 1
    Total Subsets 16
  5. fill up the table given below
Set number of elements number of subsets total subsets \(2^{n}\)
\(A\) 1 2 \(2^1 = 2\)
\(B\) 2 4 \(2^2 = 4\)
\(C\) 3 8 \(2^3 = 8\)
\(D\) 4 16 \(2^4 = 16\)


Let \( N = \{x : x \text{ is counting number up to } 5\} \). Express set \( N \) by the listing method. Make the following subsets from the given set and name them.

\(N = \{1, 2, 3, 4, 5\}\)

  1. Subset that has only one element.
    e.g., \(\{1\}\)
  2. Subset that has two elements.
    e.g., \(\{1,2\}\)
  3. Subset that has three elements.
    e.g., \(\{1,2,3\}\)
  4. Subset that has four elements.
    e.g., \(\{1,2,3,4\}\)
  5. Subset that has five elements.
    \(\{1,2,3,4,5\}\)
  6. Subset having no elements (empty set).
    \(\phi\)
  7. Write the number of subsets formed from the given set \( N \).
    \(2^5 = 32\)


How Many Subsets? Quiz

Given the set below, how many total subsets does it have?



Power Set

कुनै एक समुह \( S \) को सबै सम्भावित उपसमुहहरूको समुहलाई \( S \) को Power Set भनिन्छ। यसलाई \( P(S) \) द्वारा जनाइन्छ। जस्तै, यदि \( S = \{a, b, c\} \) भने
\( P(S) = \{\phi, \{a\}, \{b\}, \{c\}, \{a, b\}, \{b, c\}, \{a, c\}, \{a, b, c\}\} \)।
जसमा
  1. \( n(P(S)) = 2^{n(S)} \)
  2. \( S \in P(S) \)

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