# Current Affairs 11th Class

## Notes - Mathematics Olympiads - Set Theory

Category : 11th Class

Set Theory

A Set

A set is the collection of things which is well-defined. Here well-defined means that group or collection of things which is defined distinguishable and distinct e.g. Let A is the collection of the group M = {cow, ox, book, pen, man}.

Is this group collection is a set or not? Actually this is not a set. Because it is collection of things but it cannot be defined in a single definition.

For example, A = {1, 2, 3, 4, 5,... n}

Here, collection A is a set because A is group or collection of natural numbers.

e.g. A = {a, e, i, o, u}= {x/x : vowel of English alphabet}

Distinguish Between a Set and a Member

e.g.       A = {2, 5, 8, 7}

$2\in A$. It is read as 2 belongs to set A. Or 2 is the member/element of set A.

$6\notin A$- It means 6 doesn't belong to set A. A set is represented by capital letter of English/Greek Alphabet.

$\alpha ,\beta ,\gamma ,\delta$or A, B, C, U, S etc.

and its all elements is closed with { } bracket.

Hence what is the difference between 2 and {2}

$\because$2 can be element of a set whereas {2} represents a set whose one elements is 2.

Note:    A set is the collection of distinguish and distinct things

A set can be written/represented in the two form

(a)        Tabular form/Roaster form

(b)        Set builder form

e.g.

1. $A=\{a,e,i,o,u\}\to$Tabular form/roaster form

= {$x/x:$ a vowel of english letter} = It is said to be set-builder form

Note:    $x:x$or $x/x$is read as x such that $x$

1. A = {2, 4, 6, 8, 10} = {$x/x$, is an even numbers $\le$10}

Type of Sets

(i)         Null Set: A set which has no element, is said to be a null set and denoted by $\phi \,or\,\{\}$

e.g.       (a) a set of three-eyed men

(b) Set of real solution of the equation ${{x}^{2}}+1=0,$ ${{x}^{2}}=-1\Rightarrow x=\sqrt{-1}=$imaginary

(ii)        Singleton Set: A set having single element is said to be singleton set.

e.g.       A = {2}

(iii)       Finite Set: A set having definite and countable elements, is said to be finite set

e.g.       $A=\{a,e,i,o,u\}$

(iv)       Infinite Set: A set having uncountable and indefinite elements is said to be infinite set.

e.g.       a set of stars.

(v)        Equal set: Two or more than two sets are said to be equal sets if they have the same elements.

e.g.       $A=\{1,\,2,\,5,\,7,\,8\}$, $B=\{2,\,5,8,1,7\}$. So, $A=B$

• Operations on Sets
1. Union-operation
2. Intersection operation
3. Complement operation
4. Difference operation

e.g.       $A=\{1,\,2,\,3,\,5\}$, $B=\{2,\,5,7,\,8\}$

$A\,\bigcup B=\{1,\,2,3,5,7,8\}$, $A\,\bigcap \,B=\{2,\,5\}$

• Union Set: Let A and B be two non-empty sets then A union B i.e. $A\,\bigcup B$is a set whose elements are the element of set A or B or both A and B.

• Intersection Set: Let A and B be two non-empty sets. Then $A\,\bigcap \,B$ is a set whose elements are the element of both A and B set.

• Universal Set: A collection of given all set is said to be universal set generally universal set is denoted by$\,\bigcup$.

e.g.       $\mathbf{U}=\{1,\,2,3,4,\,5,6,\,7,8,\,9,\,10\}$ Universal set, $A=\{1,\,4,\,5,\,10\}$, $B=\{2,\,3,6,\,7\}$

$\therefore \,\,A-B=$difference between A and B$=\{1,\,4,\,5,\,10\}$, $\,B-A=\{2,\,3,6,7\}$

A' = complement of $A=\bigcup -A=\{2,\,3,\,6,\,7,8\}$, B' $=\mathbf{U}-B$

$=\{1,\,2,3,4,\,5,6,\,7,8,\,9,\,10\}-\{2,\,3,\,6,\,7\}=\{1,\,4,\,5,\,8,\,10\}$

$A\,\Delta \,B=$Symmetric difference between A & B $=\mathbf{(A-B)}\,\,\mathbf{U}\,\,\mathbf{(B-A)}$

$=\{1,\,4,\,5,10\}\,\,\bigcup \,\,\{2,\,3,\,6,\,7\}=\{1,\,2,3,\,4,\,5,6,7,10\}$

Venn Diagram

$U=\{1,\,2,3,4,5,6,7,8,9,\,10\}$

$A=\{2,\,3,\,4,\,6,7,8,\,10\}$

$B=\{1,3,\,5,7,\,\}$

$A\,\,\bigcup \,\,B=\{1,\,2,3,4,5,\,6,7,8,10\}$

$A\,\,\bigcap \,\,B=\{3,7\}$

• Power Set: Let A is a set. Total no. of all possible subsets of set A is denoted by${{2}^{n(A)}}$. It is said to be power set. It is represented by P(A)

e.g.       $A=\mathbf{\{2,}\,\mathbf{3,}\,\mathbf{5\}}$, $n(A)=3$. Total number of subset $={{2}^{n(A)}}={{2}^{3}}=8$

• Possible Subset: $\{\phi ,\,\{2\},\,\{3\},\,\{5\},\,\{2,\,3\},\,\{3,\,5\},\,\{2,\,5\},\,\{2,3,\,5\}\}$

• Definition (Subset): Let A & B be two non-empty sets. We say that A is subset of B. It means that all elements of set A contains in set B and it is denoted by $A\subset B$

e.g.       $A=\{2,\,4,\,5,6\}$,$B=\{1,\,2,4,\,5,\,6,7,\,8\}$, then $A\subseteq B$ is true?, $B\subseteq A$

$B\supseteq A\to B$is a subset of A

• Power Set: $A=\{2,\,3,8\}$

$P(A)=\{\phi ,\,\{2\},\{3\},\{8\},\{2,3\},\{3,8\},\{2,8\},\{2,3,\,8\}\}$

• Cartesian Product of Two sets: Let A & B be two non-empty sets then cartesian product of A & B is written as $A\times B$& it is set of ordered pair of element of two sets.

Mathematically, it is written as

$A\times B=\{(x,y);\,\,x\in A\,\And Y\in B\}$

$(x,y)\to$ Linear ordered pair

e.g.                   $A=\text{ }\!\!\{\!\!\text{ 1,}\,\text{2,}\,\text{3 }\!\!\}\!\!\text{ }$, $\text{B= }\!\!\{\!\!\text{ 2,}\,\text{5 }\!\!\}\!\!\text{ }$

$A\times B=\{1,\,2,\,3\}\times \{2,5\}=\{(1,\,2),\,(1,5),(2,\,2),(2,5),(3,\,2),\,(3,5)\}$

$B\times A=\{2,5\}\times \{1,\,2,3\}=\{(2,1),(2,2),(2,3),(5,1),(5,\,2),(5,\,3)\}$worked on

1. Find x & y of

$(3x+y,x-1)=(x+3,2y-2x)$,

$3x+y=x+3\Rightarrow 3x-x+y=3$

$\Rightarrow 2x+y=3$                                       ......... (1)

$x-1=2y-2x$

$2y-2x-x=-1\Rightarrow 2y-3x=-1$                      ....... (2)

Solving (1) & (2), we have

$2x+y=3$,$2y-3x=-1$

subtracting (2) from (1) x 2, we have

$4x+2y=6$, $2y-3x=-1$

$7x=7\Rightarrow x=\frac{7}{7}=1$

Putting the value of x in (1), we have

$2\times 1+y=3\Rightarrow y=3-2=1$

$x=1$, $y=1$

1. Q. If $A=\{x:{{x}^{2}}-5x+6=0\}$

$B=\{2,\,4\}\And C=\{4,\,5\}$

Find $(A-B)\times (B-C)$

Sol:      ${{\mathbf{x}}^{\mathbf{2}}}\mathbf{-5x+6=0}$

$\Rightarrow \,\,\,{{x}^{2}}-3x-2x+6=0$

$\Rightarrow \,\,\,x(x-3)-2(x-3)=0$

$\Rightarrow \,\,\,(x-3)(x-2)=0$

$\Rightarrow \,\,\,x-2,3$

Now,     $A=\{x:{{x}^{2}}-5x+6=0\}=\{2,3\}$

$A-B=\{2,\,3\}-\{2,\,4\}=\{3\}$

$B-C=\{2,\,4\}-\{4,\,5\}=\{2\}$

$(A-B)\times (B-C)=\{3\}\times \{2\}=\{(3,\,2)\}$

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