Introduction to Sets

Introduction to sets

1.Define what a set is?A set can be defined as a collection objects.  2.What do you call the objects inside the set?you call it ELEMENTS or MEMBERS.Moving onto how we talk to sets on slide two

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Caption: : A Collection of Objects

ListNotation method1 refer to set?

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Caption: : Example of elements in a set

The element belongs to the set inside the circle so lets call it the CIRCLE Set and nickname it as C.The element's / objects are represented by the E-SHAPED symbolSo now we can talk to the circle by calling on the elements eg.  3 E C (three is an Element C for circle)Example twoyou can write it out in list notation below and refer to it aswell.list notation:  C= {1,2,3,4,5,6,7}2 E C or 5 E = {1,2,3,4,5,6,7,} 8 ∉ C = 8 is not an Element of C So two or five(in fact 1-7 also) is an ELEMENT of C!

Set Builder Notation method2

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Caption: : How to refer to set using Set Builder notation example.

The example shows how to read and write out a seton the leftBelow are other otherways aswell good to know these NB!!!                  {0, 2, 4, 6, 8}, {x | x is an even non-negative integer less than 10}, and  {x | x ∈ Z≥, x is an even integer less than or equal to 8}..Z≥ refers to non-negative intergers(1,2,3,4....)Z refers to Integers that is positveand negative numbers inclusive (-1,-2,3,4).

How to know if a set is Equal?

You can have more than one set so lets compare them to see if they are equalLets call the 3x sets of Apples A,B andC all of them has the same amount and type of Appelsso they EQUAL So it makes sense to regard two sets as equal if they have precisely the same elements. So, for example {3, 4} = {4, 3} Another example thats Equal: {5, 7} = {5, 5, 7} Number 5 is listed twice in the set, it does not tell us anything we don’t already know. The numbers which are in {5, 7} are 5 and 7, and these are exactly the numbers that are in {5, 5, 7}. So these sets are equal.     

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Caption: : Equality Example in sets

Meet the Farther: The Empty Set

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Caption: : The set FAMILy: Hi i am the empty set

A set that has NO ELEMENTS is called an EMPTY SETOk! so lets talk to the empty set.In List notation we refer to it as = {}∅In Set builder notation we refer to it {x < 1 | x ∈ Z+}the set of all the X's less than one SUCH that X is anELEMENT of positive INTEGERS.There other ways to refer to an empty set to !

Meet the mother: The Union set

The union of sets A and B is denoted by A ∪ B, and is the set of all those elements which belong to A or to B (or to both). i.e A={1,2,3} and B={3,4,5}AuB={1,2,3,4,5} (Think poker all in) AuB is the SET of all X'S SUCH that X is an ELEMENTof A or X is an ELEMENT of B A ∪ B = {x | x ∈ A or x ∈ B}.“if x ∈ A ∪ B, then x ∈ A or x ∈ B”, and “if x ∈ A or x ∈ B, then x ∈ A ∪ B”. Combined stated of mother Union below"iff" = if and only if -> is what it means“x ∈ A ∪ B iff x ∈ A or x ∈ B”. 

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Caption: : All the elements of A an B added together

Meet Sister: Intersect set

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Caption: : An element that is both in A and B is 3 example

Two or more sets that share one or more ELEMENTSin common this where they intersect.denoted by A ∩ BAnB = The set of the X'S SUCH that  X's Element ofA and X's an element of B (KEY WORD AND) A ∩ B = {x | x ∈ A and x ∈ B}  KEY Word to remember is AND with Intersect in the SET BUILDER NOTATION.  “x ∈ A ∩ B iff x ∈ A and x ∈ B”  i.e Let A = {1, 2} and B = {0, 1},then A n B ={1}.

Meet Brother: Set difference(-)

We minus set A from B (A-B) then get the difference by removal.

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Caption: : Remove the elements A and B has in common write A only

Meet Uncle: Symmetric set Difference(+)

SO basically we add(+) the sets for A and B BUTDONT WRITE the elements in COMMON! i.e Let A = { 3, 4 } and B = { 4, 5, 6 }  A + B = { 3, 4 } + { 4, 5, 6}      ->  A+B  ={3, 5, 6}  REMEMBER OR used in SET BUILDER NOTATION. A+B = the SET of All the ELEMENTS SUCH that X's an ELEMENT of A or X's an ELEMENT of B but not BOTH A + B = { x | x ∈ A or x ∈ B, but not both.}  “x ∈ A + B iff x ∈ A or x ∈ B, but not both”. 

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Caption: : Remember we ADD the sets and Dont write the in common elements.

Meet Aunt: Disjointed Sets

Two sets A and B are called disjoint if they have no elements in common, i.e. there is not a single element that live in both A and B, i . e . A ∩ B = ∅Let A = { 1, 2, 3, 4 } and B = { 5, 6 } with U = {1, 2, 3, 4, 5, 6}.  Then we can say that A and B are disjoint. There are no elements that b e l o n g  t o  b o t h   A  a n d  B , i . e . A ∩ B = ∅ . 

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Caption: : They have NO Elements in COMMON

Meet Grandpa: The Sub Set

Example: {1, 2, 3} is a proper subset of {1, 2, 3, 4} because the element 4 is not in the first set. Notice that if A is a proper subset of B, then it is also a subset of B Another example: Let C = { ∅, {∅} } with ∅ and {∅} being the elements of C. (The element {∅} is highlighted so that one can remember that this is one element of C.) How we write a sub set see below!!A⊆B

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Caption: : NOTE ALL of A must be inB

Meet Grandma: Proper sub set

For example, if A={ 1, 3, 5 } then B={ 1, 5 } is a proper subset of A.REMEMBER reversed of subset ALL of B must be in AFor example, if A={ 1 ,3 ,5} then B={ 1 ,5 } is a proper subset of A. The set C={1, 3, 5 } is a subset of A, but it is not a proper subset of A since C=A,C=A. The set D= { 1, 4 } is not even a subset of A, since 4 is not an element of A.How we write it: A ⊂ B 

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Meet Kitty: Cardinal sets

The sum of a set or setsbasically you count the ELEMENTS in a set or SETS Let A = {1, 2, 3, 4} and B = {5, 6} be subsets of U = {1, 2, 3, 4, 5, 6},  then |A| = 4, |B| = 2 and |U| = 6,

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Caption: : YEs you start counting

Meet Bully the dog: Compliment set

Let A be a subset of a universal set U. Then the complement of A is defined as the set of all elements that belong to U but do not belong to

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Caption: : All belongs to the COMPLIMENT oF A not the A in the white circle

Meet Happy Hamster: Power Set

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Given a set A with n distinct elements, the power set of A, denoted by Ƥ (A), is the set that has as its members all the subsets of A. Let us list them: 0/, { 1 }, { 2 }, { 3 }, { 1, 2 }, { 1, 3 }, {2, 3} and {1, 2, 3}.  These are all the members of Ƥ (B). The power set of B, i.e. Ƥ (B) is therefore Ƥ (B) = { 0/, { 1 }, { 2 }, { 3 }, { 1, 2 }, { 1, 3 }, { 2, 3 }, { 1, 2, 3 } }. The cardinality of Ƥ (A) is 2n, i.e. |Ƥ (A)| = 2n.  Let B = {1, 2, 3}. Which sets are all subsets of B?

Caption: : Defined in picture example!!

Now you met the Family of Sets

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Caption: : Kitty the Cat,Bully the dog, Happy the hamster

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Caption: : Here they are now try to memorise them