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
• 
• Bildunterschrift: A Collection of Objects

ListNotation method1 refer to set?

• 
• Bildunterschrift: 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 symbol
• So now we can talk to the circle by calling on the elements eg. 3 E C (three is an Element C for circle)
• Example two
• you 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

• 
• Bildunterschrift: How to refer to set using Set Builder notation example.
• The example shows how to read and write out a set
• on the left
• Below 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 positve
• and 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 equal
• Lets call the 3x sets of Apples A,B andC
• all of them has the same amount and type of Appels
• so 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.
• 
• Bildunterschrift: Equality Example in sets

Meet the Farther: The Empty Set

• 
• Bildunterschrift: The set FAMILy: Hi i am the empty set
• A set that has NO ELEMENTS is called an EMPTY SET
• Ok! 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 an
• ELEMENT 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 ELEMENT
• of 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”.
• 
• Bildunterschrift: All the elements of A an B added together

Meet Sister: Intersect set

• 
• Bildunterschrift: An element that is both in A and B is 3 example
• Two or more sets that share one or more ELEMENTS
• in common this where they intersect.
• denoted by A ∩ B
• AnB = The set of the X'S SUCH that X's Element of
• A 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.
• 
• Bildunterschrift: 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 BUT
• DONT 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”.
• 
• Bildunterschrift: 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 = ∅
• .
• 
• Bildunterschrift: 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
• 
• Bildunterschrift: 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 A
• For 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
• 

Meet Kitty: Cardinal sets

• The sum of a set or sets
• basically 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,
• 
• Bildunterschrift: 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
• 
• Bildunterschrift: All belongs to the COMPLIMENT oF A not the A in the white circle

Meet Happy Hamster: Power Set

• 
• 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?
• Bildunterschrift: Defined in picture example!!

Now you met the Family of Sets

• 
• Bildunterschrift: Kitty the Cat,Bully the dog, Happy the hamster
• 
• Bildunterschrift: Here they are now try to memorise them