TASK TWO

# Way of listing the elements of Sets




There are two ways of writing sets:

1. Roster Method

-listing the elements in any order and enclosing them with braces.

Example:

A= {January, February, March…December}

B={1,3,5…}

2. Rule Method

-giving a descriptive phrase that will clearly identify the elements of

the set.

Example:

C={days of the week}

D={odd numbers}


#Properties of sets

-Identity properties
-Domination properties
-Idempotent properties
-Commutative properties
-Associative properties
-Distributive properties
-De morgan's properties


#Set membership

Relation “is an element of”

Symbol :  and 

Example: X  A

• X is an element of A  “X is a member of A”
• X is a subset of A

Example

A = {1, 2, 3, 4}, B = {blue, white, red}, F = {n^2 – 4: n is an integer, 0  n  19}

4  A and 12  F; but 9  F and green  B

#Empty set

Symbol(null set) : {} and 

Set with no element

Example: A = {2,6,8} and B = {3, 5, 7}

Let C is intersection A and B

Thus, C = {}, null set because no element common between the two sets

#Set of numbers 

Symbol	Description
N	Natural Numbers The whole numbers from 1 upwards. (Or from 0 upwards in some fields of mathematics).The set is {1,2,3,...} or {0,1,2,3,...}
Z	Integers The whole numbers, {1,2,3,...} negative whole numbers {..., -3,-2,-1} and zero {0}. So the set is {..., -3, -2, -1, 0, 1, 2, 3, ...} (Z is for the German "Zahlen", meaning numbers, because I is used for the set of imaginary numbers).
Q	Rational Numbers The numbers you can make by dividing one integer by another (but not dividing by zero). In other words fraction.  Q is for "quotient" (because R is used for the set of real numbers). Examples: 3/2 (=1.5), 8/4 (=2), 136/100 (=1.36), -1/1000 (=-0.001), etc.
	Irrational Numbers Any number that is not a Rational Number.
A	Algebraic Numbers Any number that is a solution to a polynomial equation with rational coefficients. Includes all Rational Numbers, and some Irrational Numbers.
	Transcendental Numbers Any number that is not an Algebraic Number Examples of transcendental numbers include π and e.
R	Real Numbers All Rational and Irrational numbers. They can also be positive, negative or zero. Includes the Algebraic Numbers and Transcendental Numbers. Examples: 1.5, -12.3, 99, √2, π
I	Imaginary Numbers Numbers that when squared give a negative result. If you square a real number you always get a positive, or zero, result. For example 2×2=4, and (-2)×(-2)=4 also, so "imaginary" numbers can seem impossible, but they are still useful! Examples: √(-9) (=3i), 6i, -5.2i The "unit" imaginary numbers is √(-1) (the square root of minus one), and its symbol is i, or sometimes j. i2 = -1
C	Complex Numbers A combination of a real and an imaginary number in the form a + bi, where a and b are real, and i is imaginary. The values a and b can be zero, so the set of real numbers and the set of imaginary numbers are subsets of the set of complex numbers. Examples: 1 + i, 2 - 6i, -5.2i, 4

#Set Equality

-Two set or equal if they contain of the same element.

Example 1 :

If, F = {20, 60, 80}
And, G = {80, 60, 20}
Then, F=G, that is both sets are equal.

Example 2 :

If, F = {2, 4, 6, 8, 10}
And, G = {10, 12, 18, 20, 22}
Then, n(F)= n(G)= 5, that is, sets F and G are equivalent.

#Venn Diagram

is a set diagram that shows all possible logical relations between a finite collection of sets.

 

        A  B

           A  B

#Subset

A portion of set.

Example :

A set N is a subset of a set X

If, X = {3, 5, 6, 8, 9, 10, 11, 13}

And, N = {5, 11, 13}

Then, N is a subset of X.

That is, N is a subset of  X 

PROPER SUBSET

-Is a subset that strictly contained in a set.

Example :

If S={1,2,3,4}

Then T={1,2,3}

T is a proper subset of S.

IMPROPER SUBSET

-Subset consist of all element of a given set.

Example :

If W ={1,2,3,4}

Then V= {1,2,3,4}

W is a improper subset of V.

#Power Set

Collection of sets which represents every valid subset of a set

Example:

F = {Apple, Orange, Grape}

Member of power set:

, {Apple} , {Orange}, {Grape}, {Apple, Orange}, {Orange, Grape}, {Apple, Grape}, {Apple, Orange, Grape}

Write as:

P (P)= , {Apple} , {Orange}, {Grape}, {Apple, Orange}, {Orange, Grape}, {Apple, Grape}, {Apple, Orange, Grape}

#Set Operation

Union
Two sets can be “added” together. The Union of A and B, denoted by A U B, is the set of all things which are members of either A or B.
Example:
 {3,4} U {A,B} = {3,4,A,B}

Properties:
A U B =B U A
A U (B U C) = (A U B) U C
A ≤ (A U B)
A ≤ B if and only if A U B = B
A U A = A
A U  = A

Intersection
The intersection of A and B, denoted by A  B, is the set of all things which are member of both A and B.
Example :
{1, 2}  {3, 4} =

Properties:
A  B = B  A
A  (B  C) = (A  B)  C
A  B  A
A  A = A
A   =
A  B if and only if A  B = A

Set Difference

The set difference between A and B is the set of all elements of A that are not in B, denoted by A/B or A-B.

Example:
{1, 2, 3} \ {2, 3, 4} = {1}

Properties:
A \  = A
A \ A =  =  \ A
B \ (A  B) = B \ A
A \ B = A  Bc
(A \ B) c  = Ac U B
(A \ B)  (C \ D) = (A  C) \ (B U D)

Disjoint set
Two set A and B are disjoint if their intersection is empty set

Example :
A  B = 

Complement of a set

Complement of A is the set of all element in the universal set U, but not in A, denoted A c

Example:
Let A = {1, 2, 3, 4}
Let U = {1, 2, 3, 4, 5, 6}
Ac  = {5, 6} 

#Generalised Union and Intersection

-Since union & intersection are commutative and associative, we can extend them from operating on ordered pairs of sets (A,B) to operating on sequences of sets (,…,), or even unordered sets of sets, X={A | Q(A)}.

Generalized Union

-Binary union operator: A∪B

- n-ary union:

∪∪…∪:≡ ((…((∪) ∪…)∪)

(grouping & order is irrelevant)

-“Big U” notation:

-Or for infinite sets of sets:

Generalized Intersection

- Binary intersection operator: A∩B

- n-ary intersection:

A∩∩…∩≡((…((∩)∩…)∩)

(grouping & order is irrelevant)

-“Big Arch” notation:

-Or for infinite sets of sets:

#Cartesian Product

Product of two different sets

Example: 

Colour = {red, black}

Car = {Ferrari, saga}

Colour X Car = {(red, ferrari), (red, saga), (black, ferrari), (black, saga)}

Colour X Car  Car X Colour

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