Set

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SET

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SETS

A group of objects A collection of distinct objects,

considered as an object in its own right.

Sets are one of the most fundamental concepts in mathematics

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SETS

Example:i. Set of even numbers less than 10 2,4,6,8ii. Set of vowels : a,e ,i,o,uiii. Colours of rainbow : red, orange,

yellow, green, blue, indigo and violet

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SETS

Can be defined by * description * using set notation,

e.g i. A set of students of WMS ii. 2,3,5,7

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Sets are named with capital letters and the objects are enclosed in braces

e.g : Set A is a set of the first seven

numbers in multiple of 7 ( description)

A = 7,14,21,28,35,42,49 (set notation)

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The objects are known as elements of the set.

set notation /symbol : : is not an element of A = 7,14,21,28,35,42,49 21 is an element of set A 21 A 8 A

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Set also can be represented by a Venn diagram

Venn Diagram: an enclosed geometrical figure, i.e circle, triangle, rectangle

e.g P = factors of 9= 1,3,9

P. 3

.9 .1

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n ( A ) means the number of elements of set A.

e.g i. P = 1,4,9,16 ,25 n ( P ) = 5 ii. Q = AVATAR Q = A, V, T ,R n (Q) = 4

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P . 3 P =

3

P 3 n( P )

= 3

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Empty Sets : A set which does not have any elements

Symbol : or

e.g Set of odd numbers which can be divided by 2.

=

e.g A = A polygon with two sides

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Equal sets:

Consider sets A= 1,2,3,4 and B = 1,4,2,3 . All elements of set A are elements of set B , hence set A is equal to set B.

A = B

e.g K = 11,12,13,14 and L = 11,14,12 The element 13 is not found in set L,

therefore, K and L are not equal sets. K L

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e.g Given that A = prime numbers less than 15 B = 2, 3, 5, 11, 13 , y +2 and A = B, find the value of y.

Solution : A = 2, 3, 5, 7, 11, 13 y + 2 = 7 y = 5

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SUBSET

Set P is a subset of set Q if all the elements of set P are also found in set Q.

“ Set P is subset of set Q” and is written as P Q“ set P is not a subset of set Q” is

denoted by P Q

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e.g Given that A = even numbers from 2 to 16 B = 4, 16 C = perfect squares less than

20. State whether each of the following is true or false.

a) B A b) B C c) C A d) A B

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Venn Diagram : P Q

P

Q

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P = 5, 6, 10 and Q = 6,10 .

P Q .10 . 5 .6

Q P

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Listing the subsets of a set .

i. P = 2 Subsets of P : , 2ii. P = 2, 4 Subsets of P : , 2 4 2,4iii. P = 2,4,6 Subsets of P = , 2 4 ,6 2,4

2,6 4,6 ,2,4,6

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# 1 : an empty set is a subset of any set

#2 : A set is a subset of itself

#3 : number of subsets of set P is given by

2)(An

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Universal set.

- the set that contains all the elements under discussion and is denoted by the symbol

.

- any set under discussion is a subset of the universal set.

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e.g Given that = x: 4 x 10, x is an integer

P = x: x is a multiple of 3

a. List all the elements of and P.

b. Draw a Venn diagram to represent and P.

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Complement set :

The complement , A’ , of set A is a set which contains all elements in the universal set, , which are not elements of set A.

e.g = 2,3,4,5,6,7,8 A = 3,5,7,8 A’ = 2, 4,6,

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A

A’

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OPERATIONS ON SETS

Intersection of two sets.

The intersection of two sets , A and B, is the set of all the common elements of A and B, and is denoted as

A B

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Intersection of two sets

The intersection of two sets , A and B, is the set of all the common elements of A and B, and is denoted as

A B

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A B

A B

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e.g Given P =3, 4, 5, 7 , Q= 4,5, 7, 8 and

R = 6, 8 , 9. Find the following set;

a. P Q b. Q R c. P R

Solution: a. 4,5,7 b. 8 c. or

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Relationship between A B and A or B.

( A B ) A and ( A B ) B

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P

Q R

Shade the region that represents P Q R’

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e.g In a class of 40 students, 22 of them play badminton while 24 play football.Among these students,14 students play both badminton and football.

a. Draw a Venn diagram to represent the information above

b. Find the number of students who

play neither badminton nor football.

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Union of sets.

- The union of two sets , A and B is a set that consists all the elements found in the sets A and B.

- Denoted by A B ( A union B)

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e.g A = 2,4,6,8 and B = 4,8,10

A B = 2,4,6,8,10 n( A B )= 5

- A (A B ) and B (A B )

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A computer centre has 120 students. 76 of them study Java while 60 of them study Pascal. Given that 35 of them study both.

Find the number of students who study Java or Pascal.