Maths Project SETS Submitted To : Joseph Sir
Set   and its notation <ul><li>{1, 2, 3} is the set containing “1” and “2” and “3.” </li></ul><ul><li>{1, 1, 2, 3, 3} = {1...
Subsets <ul><li>x      S  means “ x  is an element of set  S .” </li></ul><ul><li>x      S  means “ x  is not an element...
Superset <ul><li>A      B  means “ A  is a subset of  B .” </li></ul><ul><li>A      B  means “ A  is a superset of  B .”...
<ul><li>A      B  means “ A  is a proper subset of  B .” </li></ul><ul><ul><li>A      B , and  A      B . </li></ul></u...
Power sets <ul><li>If  S  is a set, then the  power set  of  S  is  </li></ul><ul><li>P ( S )  = 2 S   = {  x  :  x      ...
Union <ul><li>The  union   of two sets  A  and  B  is: </li></ul><ul><li>A      B  = {  x  :  x      A      x      B }...
Intersection <ul><li>The  intersection  of two sets  A  and  B  is: </li></ul><ul><li>A     B = {  x  :  x      A      ...
Complement <ul><li>The  complement  of a set  A  is: </li></ul>If  A  = { x  :  x  is not shaded}, then <ul><li>=  U  and ...
Difference <ul><li>The  symmetric difference ,  A      B , is: </li></ul><ul><li>A      B  = {  x  : ( x      A      x...
<ul><li>Properties of the union operation: </li></ul><ul><ul><li>A U    = A Identity law </li></ul></ul><ul><ul><li>A U  ...
<ul><li>Properties of the intersection operation : </li></ul><ul><ul><li>A ∩  U  = A Identity law </li></ul></ul><ul><ul><...
Some Properties of Complement Sets 1.  Complement Laws: A U A’= U  A ∩ A’=   2.  De Morgan’s Law: (A U B)’ = A’ ∩ B’ (A ∩...
THANK YOU Done By : Sukesh XI- B
Upcoming SlideShare
Loading in...5
×

Sets

1,689

Published on

chapter : sets

0 Comments
3 Likes
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total Views
1,689
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
99
Comments
0
Likes
3
Embeds 0
No embeds

No notes for slide

Sets

  1. 1. Maths Project SETS Submitted To : Joseph Sir
  2. 2. Set and its notation <ul><li>{1, 2, 3} is the set containing “1” and “2” and “3.” </li></ul><ul><li>{1, 1, 2, 3, 3} = {1, 2, 3} since repetition is irrelevant. </li></ul><ul><li>{1, 2, 3} = {3, 2, 1} since sets are unordered. </li></ul><ul><li>{0,1, 2, 3, …} is a way we denote an infinite set (in this case, the natural numbers). </li></ul><ul><li> = {} is the empty set, or the set containing no element. </li></ul>. Introduction : A set is an unordered collection of elements. Examples. Note:   {  }
  3. 3. Subsets <ul><li>x  S means “ x is an element of set S .” </li></ul><ul><li>x  S means “ x is not an element of set S .” </li></ul><ul><li>A  B means “ A is a subset of B .” </li></ul>or, “ B contains A .” or, “every element of A is also in B .” or,  x (( x  A )  ( x  B )). Venn Diagram B A
  4. 4. Superset <ul><li>A  B means “ A is a subset of B .” </li></ul><ul><li>A  B means “ A is a superset of B .” </li></ul><ul><li>A = B if and only if A and B have exactly the same elements </li></ul>iff, A  B and B  A iff, A  B and A  B iff,  x (( x  A )  ( x  B )). <ul><li>So to show equality of sets A and B , show: </li></ul><ul><ul><li>A  B </li></ul></ul><ul><ul><li>B  A </li></ul></ul>
  5. 5. <ul><li>A  B means “ A is a proper subset of B .” </li></ul><ul><ul><li>A  B , and A  B . </li></ul></ul><ul><ul><li> x (( x  A )  ( x  B )) </li></ul></ul><ul><ul><li>  x (( x  B )  ( x  A )) </li></ul></ul>A B
  6. 6. Power sets <ul><li>If S is a set, then the power set of S is </li></ul><ul><li>P ( S ) = 2 S = { x : x  S }. </li></ul>If S = { a } If S = { a,b } If S =  If S = {  ,{  }} We say, “ P ( S ) is the set of all subsets of S .” 2 S = {  , { a }}. 2 S = {  , {a}, {b}, {a,b}}. 2 S = {  }. 2 S = {  , {  }, {{  }}, {  ,{  }}}.
  7. 7. Union <ul><li>The union of two sets A and B is: </li></ul><ul><li>A  B = { x : x  A  x  B } </li></ul>If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi}, then A  B = {Charlie, Lucy, Linus, Desi} A B
  8. 8. Intersection <ul><li>The intersection of two sets A and B is: </li></ul><ul><li>A  B = { x : x  A  x  B } </li></ul>If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi}, then A  B = {Lucy} A B
  9. 9. Complement <ul><li>The complement of a set A is: </li></ul>If A = { x : x is not shaded}, then <ul><li>= U and </li></ul><ul><li>U =  </li></ul>A U
  10. 10. Difference <ul><li>The symmetric difference , A  B , is: </li></ul><ul><li>A  B = { x : ( x  A  x  B )  ( x  B  x  A )} </li></ul>= ( A – B )  ( B – A ) = { x : x  A  x  B } U A – B B – A
  11. 11. <ul><li>Properties of the union operation: </li></ul><ul><ul><li>A U  = A Identity law </li></ul></ul><ul><ul><li>A U U = U Domination law </li></ul></ul><ul><ul><li>A U A = A Idempotent law </li></ul></ul><ul><ul><li>A U B = B U A Commutative law </li></ul></ul><ul><ul><li>A U (B U C) = (A U B) U C Associative law </li></ul></ul>
  12. 12. <ul><li>Properties of the intersection operation : </li></ul><ul><ul><li>A ∩ U = A Identity law </li></ul></ul><ul><ul><li>A ∩  =  Domination law </li></ul></ul><ul><ul><li>A ∩ A = A Idempotent law </li></ul></ul><ul><ul><li>A ∩ B = B ∩ A Commutative law </li></ul></ul><ul><ul><li>A ∩ (B ∩ C) = (A ∩ B) ∩ C Associative law </li></ul></ul>
  13. 13. Some Properties of Complement Sets 1. Complement Laws: A U A’= U A ∩ A’=  2. De Morgan’s Law: (A U B)’ = A’ ∩ B’ (A ∩ B )’ = A’ U B’ 3. Law of Double Complementation: ( A’ )’ = A 4. Laws of Empty Set and Universal Set:  ’ = U U’ = 
  14. 14. THANK YOU Done By : Sukesh XI- B
  1. A particular slide catching your eye?

    Clipping is a handy way to collect important slides you want to go back to later.

×