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# Sets

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chapter : sets

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### 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
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