11th Class Mathematics Sets Notes - Mathematics Olympiads - Set Theory

 

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 \[x\]

 

2. 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,     \[A=\{x:{{x}^{2}}-5x+6=0\}=\{2,3\}\]

\[A-B=\{2,\,3\}-\{2,\,4\}=\{3\}\]

\[B-C=\{2,\,4\}-\{4,\,5\}=\{2\}\]

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