mathematics-class-11-unit-01-chapter-01-set-theory-l-1-3-iqo1nw-wsni

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-1 </B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B> Set </B></Primary></p>
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<p><strong>Definition</strong></p>
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<p>A set is a well defined Collection of objects (or elements).</p>
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<p>By well-defined collection, we mean that given an object we can clearly determine whether the object is in the collection or not.</p>
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<span style="color:blue">Set</span> $\to$ Example   $\to$  Notation  $\to$ Representation of sets   $\to$  2. Set-builder form  $\to$  Some notations of sets  $\to$ Empty set $\to$ Finite & infinite sets $\to$ Equal sets $\to$ Subsets $\to$ Power sets$\to$ Cardinality of power set $\to$ Operation on sets $\to$ Singleton set $\to$ Complement of a set  
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-1 </B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Notation</B></Primary></p>
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<p>We shall usually denote sets by capital letters, e.g; A,B,C,X,Y etc.</p>
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<p>We denote the elements by small letters like a,b,c,x,y,z,etc.</p>
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<p><strong>Notation</strong>: We write “a∈A” (read as “a belongs to A” or “a is an element of A”) to say a is an element of the set A.</p>
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<p>e.g. Let A be the set of all even natural numbers. Then 2∈A but 3∉A.</p>
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Set $\to$ Example   $\to$ <span style="color:blue"> Notation </span> $\to$ Representation of sets   $\to$  2. Set-builder form  $\to$  Some notations of sets  $\to$ Empty set $\to$ Finite & infinite sets $\to$ Equal sets $\to$ Subsets $\to$ Power sets$\to$ Cardinality of power set $\to$ Operation on sets $\to$ Singleton set $\to$ Complement of a set  
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-1 </B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>2. Set-builder form</B></Primary></p>
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<p>The set is described by the characteristics property possessed by all the elements of the set.</p>
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<p>e.g. The set A={2,4,6,8,10} is represented in set-builder form as A={n:n is a natural number divisible by 2 and n≤10}.</p>
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Set $\to$ Example   $\to$  Notation  $\to$ Representation of sets   $\to$ <span style="color:blue"> 2. Set-builder form</span>  $\to$  Some notations of sets  $\to$ Empty set $\to$ Finite & infinite sets $\to$ Equal sets $\to$ Subsets $\to$ Power sets$\to$ Cardinality of power set $\to$ Operation on sets $\to$ Singleton set $\to$ Complement of a set  
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-1 </B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B> Some notations of sets</B></Primary></p>
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<li>Some notations of sets used in Mathematics:
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<li>$\N$ : set of all natural numbers.</li>
<li>$\Z$ : set of all integers.</li>
<li>$\mathbb{Q}$ : set of all rational numbers.</li>
<li>$\R$ : set of all real numbers.</li>
<li>$\mathbb{C}$ : set of all complex numbers.</li>
<li>$\R^+$ : set of all positive real numbers.</li>
<li>$\mathbb{Q}^+$ : set of all positive rationals.</li>
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Set $\to$ Example   $\to$  Notation  $\to$ Representation of sets   $\to$  2. Set-builder form  $\to$ <span style="color:blue"> Some notations of sets </span> $\to$ Empty set $\to$ Finite & infinite sets $\to$ Equal sets $\to$ Subsets $\to$ Power sets$\to$ Cardinality of power set $\to$ Operation on sets $\to$ Singleton set $\to$ Complement of a set  
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-1 </B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Empty set</B></Primary></p>
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<p>The set containing no elements.</p>
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<p>Notation $: \phi,\{\}$.</p>
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<p>The empty set is also called the “null set” or the “void set”.</p>
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<p>e.g. $A=\{n \in \mathbb{N}: 1<n<2\}$</p>
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<p>This set is empty because there is no natural number strictly between $1 \& 2$.</p>
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<li>$\mathrm{B}=\left\{x \in \mathbb{Q}: x^2=2\right\}$</li>
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Set $\to$ Example   $\to$  Notation  $\to$ Representation of sets   $\to$  2. Set-builder form  $\to$  Some notations of sets  $\to$ <span style="color:blue">Empty set</span> $\to$ Finite & infinite sets $\to$ Equal sets $\to$ Subsets $\to$ Power sets$\to$ Cardinality of power set $\to$ Operation on sets $\to$ Singleton set $\to$ Complement of a set  
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-1 </B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Finite & infinite sets</B></Primary></p>
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<p>A set is called a <strong>finite set</strong> if it contains only finitely many elements; otherwise it is called an <strong>infinite set</strong>.</p>
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<p>e.g.</p>
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<li>A = set of all vowels is a finite set.</li>
<li>$\N$, $\Z$, $\mathbb{Q}$, $\R$, $\mathbb{C}$ are infinite.</li>
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Set $\to$ Example   $\to$  Notation  $\to$ Representation of sets   $\to$  2. Set-builder form  $\to$  Some notations of sets  $\to$ Empty set $\to$<span style="color:blue"> Finite & infinite sets </span>$\to$ Equal sets $\to$ Subsets $\to$ Power sets$\to$ Cardinality of power set $\to$ Operation on sets $\to$ Singleton set $\to$ Complement of a set  
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-1 </B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Equal sets</B></Primary></p>
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<p>Two sets A and B are said to be equal if they contain exactly the same elements.</p>
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<p>Note that while representing a set the order of the elements is not important.</p>
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<p>For example: If<br>
A={1,2,3} & B={2,3,1}.<br>
Then A = B.</p>
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Set $\to$ Example   $\to$  Notation  $\to$ Representation of sets   $\to$  2. Set-builder form  $\to$  Some notations of sets  $\to$ Empty set $\to$ Finite & infinite sets $\to$<span style="color:blue"> Equal sets</span> $\to$ Subsets $\to$ Power sets$\to$ Cardinality of power set $\to$ Operation on sets $\to$ Singleton set $\to$ Complement of a set  
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-1 </B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Subsets</B></Primary></p>
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<li>Let $X$ be a set.</li>
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<p>We say that a set $A$ is a Subset of $X($ denoted by $A \subseteq X)$ if every elements of $A$ is also an element of $X$. $A \subseteq X$ if $a \in A $$\Rightarrow$$ a \in X$</p>
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<p>Note that:</p>
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<p>$\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R} \subseteq \mathbb{C}$.</p>
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<p>Two sets $A$ and $B$ are equal if and only if $A \subseteq B$ and $B \subseteq A$. $A=B<=>A \subseteq B$ and $B \subseteq A$</p>
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Set $\to$ Example   $\to$  Notation  $\to$ Representation of sets   $\to$  2. Set-builder form  $\to$  Some notations of sets  $\to$ Empty set $\to$ Finite & infinite sets $\to$ Equal sets $\to$<span style="color:blue"> Subsets</span> $\to$ Power sets$\to$ Cardinality of power set $\to$ Operation on sets $\to$ Singleton set $\to$ Complement of a set  
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-1 </B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Power sets</B></Primary></p>
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<p>Given a set A, the power set of A is the collection of all subsets of A.</p>
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<p><strong>Notation</strong>: P(A) denotes the power set of A, so<br>
P(A)={B:B⊆A}</p>
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<p>Let A={1,2,3}<br>
Then P(A)={φ,{1},{2},{3},{1,2},{2,3},{1,3},{1,2,3}}.<br>
Here the number of elements in P(A)=8=2<sup>3</sup>.</p>
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<p>For a set A, we denote by |A| the number of elements in A</p>
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Set $\to$ Example   $\to$  Notation  $\to$ Representation of sets   $\to$  2. Set-builder form  $\to$  Some notations of sets  $\to$ Empty set $\to$ Finite & infinite sets $\to$ Equal sets $\to$ Subsets $\to$<span style="color:blue"> Power sets</span>$\to$ Cardinality of power set $\to$ Operation on sets $\to$ Singleton set $\to$ Complement of a set  
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-1 </B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Cardinality of power set</B></Primary></p>
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<p>If |A|=n, then |P(A)|=2<sup>n</sup>. Why?</p>
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<p>If B is any subset of A, then any particular elements of A is either in B or not in B.<br>
There are n elements and for each we get a subset by specifying whether it is in B or not.     <br>
$\Rightarrow$|P(A)|=2<sup>n</sup>.</p>
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Set $\to$ Example   $\to$  Notation  $\to$ Representation of sets   $\to$  2. Set-builder form  $\to$  Some notations of sets  $\to$ Empty set $\to$ Finite & infinite sets $\to$ Equal sets $\to$ Subsets $\to$ Power sets$\to$ <span style="color:blue">Cardinality of power set</span> $\to$ Operation on sets $\to$ Singleton set $\to$ Complement of a set  
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-1 </B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Operation on sets</B></Primary></p>
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<p>(2) $\underline{\text{Intersection:}}$</p>
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<p>A∩B = {$x$:$x$∈A and $x$∈B} all elements which are common to A & B.</p>
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<p>e.g. A={1,2,3}, B={2,3,4,5}  <br>
A∩B={2,3}</p>
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<p>If A∩B = φ then we say that A & B are disjoint</p>
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Set $\to$ Example   $\to$  Notation  $\to$ Representation of sets   $\to$  2. Set-builder form  $\to$  Some notations of sets  $\to$ Empty set $\to$ Finite & infinite sets $\to$ Equal sets $\to$ Subsets $\to$ Power sets$\to$ Cardinality of power set $\to$ <span style="color:blue">Operation on sets</span> $\to$ Singleton set $\to$ Complement of a set  
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-1 </B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Operation on sets</B></Primary></p>
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<p>(3) $\underline{\text{Set difference:}}$</p>
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<p><strong>Notation</strong>: A\B or A$-$B</p>
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<p>A\B = {$x$ : $x$∈A and $x$∉B}</p>
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<p>e.g. A = {1,2,3}, B = {2,3,4,5}<br>
A\B = {1} , B\A = {4,5}</p>
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Set $\to$ Example   $\to$  Notation  $\to$ Representation of sets   $\to$  2. Set-builder form  $\to$  Some notations of sets  $\to$ Empty set $\to$ Finite & infinite sets $\to$ Equal sets $\to$ Subsets $\to$ Power sets$\to$ Cardinality of power set $\to$ <span style="color:blue">Operation on sets</span> $\to$ Singleton set $\to$ Complement of a set  
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-1 </B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Complement of a set</B></Primary></p>
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<p>Let ∪ be an universal set & A be a subset of ∪.  <br>
The complement of A in U is A’ = U\A .</p>
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<p>Complement of A consist of all elements which are not in A.</p>
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<p>e.g:-</p>
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<li>∪={1,2,3,4}, A={2,3}<br>
A’={1,4}=U\A</li>
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Set $\to$ Example   $\to$  Notation  $\to$ Representation of sets   $\to$  2. Set-builder form  $\to$  Some notations of sets  $\to$ Empty set $\to$ Finite & infinite sets $\to$ Equal sets $\to$ Subsets $\to$ Power sets$\to$ Cardinality of power set $\to$ Operation on sets $\to$ Singleton set $\to$ <span style="color:blue">Complement of a set  </span>
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