g8_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 all possible 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 are
    {} and \(\{1, 5, 8, 9, 10\}\)
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 \(A=\{a\}, B=\{a,b\}, C=\{a,b,c\}, D=\{a,b,c,d\}\), 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

fill up the table given below

Complete the table by filling in the number of subsets for each set.

Set Number of elements (n) Number of subsets Total subsets \(2^{n}\)
\(A\) 1 \(2^1 = ?\)
\(B\) 2 \(2^2 = ?\)
\(C\) 3 \(2^3 = ?\)
\(D\) 4 \(2^4 = ?\)

Test your understanding

Given the set A = {a, e, i}, answer the following questions:

  1. How many subsets of 1 element?
  2. How many subsets of 2 elements?
  3. How many subsets of 3 elements?

Test your understanding

Given the set A = {a, e, i, o}, answer the following questions:

  1. How many subsets of 1 element?
  2. How many subsets of 2 elements?
  3. How many subsets of 3 elements?
  4. How many subsets of 4 elements?

Test your understanding

Given the set A = {a, e, i, o, u}, answer the following questions:

  1. How many subsets of 1 element?
  2. How many subsets of 2 elements?
  3. How many subsets of 3 elements?
  4. How many subsets of 4 elements?



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., \(\{a,e,i,o\}, \{a,e,i,u\}, \dots\))


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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