mathematics-set theory

8/13/2019 mathematics-set theory

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S T TH ORY

Basic Concept of Sets Venn DiagramSet Operations and Applications


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ASET IS A WELL DEFINED COLLECTION OF DISTINCT OBJECTS.

A N ELEMENT, OR MEMBER, OF A SET IS ANY ONE OF THE DISTINCT OBJECTS THAT MAKE UP THAT SET.


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Roster/Tabular Method

Rule/Textual Method

A = {1,2,3,4,5,6,7,8,9}

A={x x is a counting number from 1 to 9}

Two Methods in Writing a Set

B = {red, orange, yellow, green, blue, indigo, violet}

B={x x is a rainbow color}

C = {Dinalupihan, Hermosa, Orani, Samal,Abucay}

C={x x is a town on the First District of Bataan}


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Kinds of Sets

1.Universal Set (U)

2.Empty Set ({ } , )

3.Finite Set

4.Infinite Set

ex. English Alphabet U = {a,b,c,d,e .. w,x,y,z}

ex. A={a,b,c,d} B = {1,2,3,4,5}Set of common elements between A and B= { } or

ex. Set of English Alphabet

ex. Set of Whole Numbers


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Kinds of Sets

5.Equivalent Sets ( )

6.Equal Sets (=)

7.Joint Sets

8.Disjoint Sets

9.Subsets ( ) or _

ex. A={1,3,5,7,9} B = {2,4,6,8,10}5 elements 5 elements A B or B A

ex. C={a,b,c,d,e} D = {a,b,c,d,e}C = D or D = C

ex. E={4,5,6,7,8} F = {3,4,5,6,7}E and F are Joint Sets

ex. A={1,3,5,7,9} B = {2,4,6,8,10} A and B are Disjoint Sets

ex. G={1,2,3,4,5,6,7,8,9} H = {2,3,4,5,6}H G G H

( )


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Kinds of Sets

1.Universal Set (U)

2.Empty Set ({ } , )

3.Finite Set

4.Infinite Set

5.Equivalent Sets ( )

6.Equal Sets (=)

7.Joint Sets8.Disjoint Sets

9.Subsets ( ) or _( )

-is the totality of elements under consideration.

-set with no element.

-set with countable elements.

-set with uncountable elements.

-two sets with the same number of elements.

-two sets with exactly the same elements.

-two sets with common elements.

-two sets without common elements.

-portion of a larger set.


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1.Union of Sets ( )

Operations on Sets

2.Intersection of Sets ( )

A={1,2,3,4,5,6} B={2,4,6,8,10}

{1,2,3,4,5,6,8,10} A B=

A={1,2,3,4,5,6} B={2,4,6,8,10}

A B= {2,4,6} C={1,3,5,7,9}

A C= {1,3,5}

B C= { }


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Operations on Sets

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

{1,7,8,9,10}

A = {2,3,4,5,6} B = {1,2,3,5,6,7,9,10}

A={1,2,3,4,5,6} B={2,4,6,8,10}

A / B= {1,3,5} C={1,3,5,7,9}

A C= {2,4,6}

B/A= {8,10}

3.Complement of a Set ( c )( )

A or Ac =

B or B

c =

{4,8}

4.Difference of two Sets ( / )( )

C A= {7,9}


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Examples on Operations on Sets

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

A = {2,3,4,5,6} B = {1,3,5,7,9}

= {1,2,4,6,7,8,9,10}

C = {2,4,6,8,10}

(3,5)

= {2,3,4,5,6}

(2,4,6)

={1,2,3,4,5,6,7,9}

1) (A B)C = C

2) (A B) (A C) =(3,5) (2,4,6)

3) (A C) C C =(1,3,5,7,9)


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1.Union of Sets ( )

VENN DIAGRAM


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2.Intersection of Sets ( )

VENN DIAGRAM


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3.Complement of a Set ( )or ( c ) VENN DIAGRAM


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4.Difference of a Set ( / ) or ( ) VENN DIAGRAM


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Application:In a survey involving 150 different factories, it was

found out that:70 purchased brand A;75 purchased brand B;95 purchased brand C;30 purchased brands A and B;45 purchased brands A and C;40 purchased brands B and C;

10 purchased brands A, B and C.


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How many of them purchased:1.Brand A?

2.Brand B only?3.Brand C?4.Brands A and B?5.Brands A and B but not C?

6.Brand A or C?7.Brand A or C but not B?8.exactly 2 brands?9.at most 2 brands?10.at least 2 brands?11.all?12.How many of them did not purchased any?


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In an excursion at Pagsanjan Falls, 80 studentsbrought sandwiches, drinks and cans as follows:

50 students brought sandwiches30 students brought drinks30 students brought cans18 students brought cans and drinks15 students brought sandwiches and cans8 students brought sandwiches and drinks5 students brought sandwiches, drinks and cans.


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How many students brought:1. Sandwiches?2. Cans?3. Sandwiches only?4. At least 2 items?5. exactly 2 items?6. Sandwiches and Cans?

7. Drinks and Cans but not Sandwiches?8. exactly 1 item only?9. Cans or Sandwiches?10. Cans or Sandwiches but not Drinks?

11. Sandwiches and Cans but not Drinks?12. at most 3 items.13. all items?14. Cans and Drinks?15.how many did not bring any?

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