9th New book Ex3.1
9th New book Ex3.1 tells us how to write a set in three different ways. It also tell us about different types of a set, especially subsets and power set.
Q1 Write the following sets in set builder notation:
General form:
$$\{\;x\in S\vert condition\;on\;x\}$$
x represents the variable and S represents the set. Condition tells about the properties of the elements.
(i) {1,4,9,16,25,….,484}
$$\{x\vert x=n^2,n\in\mathbb{N}\wedge1\leq n\leq22\}$$
We can also write it as
$$\{x\in\mathbb{N}\;\vert\;x=n^2\wedge1\leq n\leq22\}$$
(ii) {2,4,8,16,32….150}
$$\{x\vert\;x=2^n,n\in\mathbb{N}\;\wedge x\leq150\}$$
{2,4,8,16,32….256}
$$\{x\vert\;x=2^n,n\in\mathbb{N}\;\wedge1\leq n\leq8\}$$
(iii) {-1000,……..-1,0,1,………..,1000}
$$\{x\;\vert\;x\in\mathbb{Z}\;\wedge-1000\leq x\leq1000\}$$
(iv) {6,12,18,….,120}
$$\{x\;\vert\;x\in\mathbb{N}\;\wedge x=6n,-1\leq n\leq20\}$$
(v) {100,102,104,….,400}
$$\{x\vert x\in\boldsymbol E\boldsymbol{\mathit\;}\mathit\wedge\mathit\;\mathit{100}\mathit\;\mathit\leq\mathit\;x\mathit\;\mathit\leq\mathit\;\mathit{400}\mathit\}$$
(vi) {1,3,9,27,81……}
$$\{x\vert x=3^n,n\in\boldsymbol W\boldsymbol{\mathit\;}\mathit\wedge\mathit\;\mathit0\mathit\;\mathit\leq\mathit\;n\mathit\}$$
(vii) {1,2,4,5,10,25,50,100}
$$\{x\;\vert\;x\in\mathbb{N}\;\wedge\;x\;divides\;100\boldsymbol{\mathit\;}\mathit\}$$
We can also write as {x | x is a factor of 100 }
(viii) {5,10,15,….,100}
$$\{x\vert\;x=5n,\;n\in\mathbb{N}\;\wedge\;1\;\leq\;n\;\leq\;20\;\}$$
(ix) The set of all integers from -100 to 100
$$\{x\;\vert\;x\in\mathbb{Z}\;\wedge-100\leq x\leq100\}$$
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Q 2 Write each of the following sets in the tabular form:
(i) $$\{x\vert\;x\;is\;a\;multiple\;of\;3\;\wedge\;x\;\leq\;35\;\}$$
{3,6,9,12,15,18,21,24,27,30,33}
(ii) $$\{x\vert\;x\in\mathbb{R}\;\;\wedge\;2x\;+1=0\;\}$$
2x+1=0
2x=-1
x= -1/2
Hence in tabular form it is {-1/2}
(iii) $$\{x\vert\;x\in\mathbb{P}\;\wedge\;x<\;12\;\}$$
{2,3,5,7,11}
(iv)
{ x | x is a divisor of 128 }
{1,2,4,8,16,32,64,128}
(v) $$\{x\vert\;x=2^n,n\in\mathbb{N}\wedge n<8\}$$
{2,4,8,16,32.64.128}
(vi) $$\{x\vert\;x\in\mathbb{N}\wedge\;x+4\;=\;0\}$$
x+4=0
x=-4 But -4 is not a natural number so set is empty and in tabular form it can be written as { }
(vii) $$\{x\vert\;x\in\mathbb{N}\wedge\;x=x\}$$
{1,2,3,4……………..}
(viii) $$\{x\vert\;x\in\mathbb{Z}\;\wedge\;3x+1\;=\;0\}$$
3x+1 = 0
3x= -1
x = -1/3 But -1/3 is not an integer hence set is empty So in tabular form we will write { }.
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Q3 Write two proper subset of each of the following set:
(i) :- {a, b, c}
{ } , {a} , {b} , {c} ,{a , b} , {a , c} , {b , c} , {a , b , c}
(ii) :- {0 , 1}
{ } , {o} , {1} , {0 , 1}
(iii) :- N
{ } , {1}
(iv) :- Z
{ } , {1}
(v) :- Q
{ } , {1}
(vi) :- R
{ } , {1}
(vii) :- $$\{x\vert x\in\mathbb{Q}\;\wedge0<x\leq2\}$$
{ } , {1}
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Q4 ;- Is there any set which has no proper subset? If so , name that set.
Yes , empty set has no proper subset.
Say A = { }
P (A) = { }
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Q5 :- what is the difference between {a , b} and {(a , b)}?
{ a , b} has two elements ‘a’ and ‘b’.
{(a , b)} has only one element which is {a , b}
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Q6 :- What is the number of element of the power set of each of the following set?
(i) :- { }
Number of element of power set of { } $$2^0\;=\;1$$
(ii) :- {0 , 1}
Number of element of power set {0 , 1} = $$2^2\;=\;4$$
(iii) :- {1 , 2 , 3 , 4 , 5 , 6 , 7}
Number of element of power set {1 , 2 , 3 , 4 , 5 , 6 , 7} = $$2^7\;=\;128$$
(iv) :- {a, {b , c}}
Number of element of power set {a, {b , c}} = $$2^2\;=\;4$$
(v) :- { {a , b} , { b , c} , {d , e} }
Number of element of power set { {a , b} , { b , c} , {d , e} } = $$2^3\;=\;8$$
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Q7 Write down the power set of each of the following sets:
(i)
A = { 9 , 11 }
P(A) = { { } , {9} , {11} , { 9 , 11}}
(ii)
$$B=\;\{\;+\;,\;-,\;\times,\;\div\}$$
$$P\;(B)\;=\;\{\;\{\;\},\{\;+\;\}\;,\;\{-\}\;,\;\{\times\}\;,\;\{\div\}\;,\\\{+\;,\;-\}\;,\;\{+,\;\times\}\;,\;\{+\;,\;\div\}\;,\;\{-\;,\;\times\}\;,\\\{-\;,\;\div\}\;,\;\{\times,\;\div\}\;,\;\{+\;,\;-\;,\;\times\}\;,\\\{\;+\;,\;\times\;,\;\div\}\;,\;\{+,-,\div\},\;\{\;-\;,\;\times,\;\div\;\},\\\{\;+\;,\;-\;,\;\times\;,\;\div\;\}\;\}$$
(iii)
$$C\;=\;\{\;\phi\;\}$$
$$P(C)\;=\{\;\{\;\}\;,\;\{\;\phi\;\}\;\}$$
$$P(C)\;=\{\;\{\;\}\;,\;\{\;\phi\;\}\;\}\;=\;\{\;\{\;\}\;,\;\{\;\{\;\}\;\}\;\}$$
(iv)
D = { a , {b , c} }
P (D) = { { } , {a} , { {b , c} } , { a , {b , c} }
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