sets.pptx

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Language of Sets

Language of Sets

_{A set is a structure, representing an unordered} collection (group, plurality) of zero or more

distinct (different) objects called elements. _{Set theory deals with operations between,}

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Basic notations for sets

 For sets, we’ll use variables S, T, U, …

 We can denote a set S in writing by listing all of its elements in curly braces:

{a, b, c} is the set of whatever 3 objects are denoted by a, b, c.

{1,2,3,4} is the set with integers from 1 to 4.

 This is possible when the set contains finite number of elements.

Basic notations for sets

 If a set is a large finite set or an infinite set, we use a descriptive notation by listing a property

necessary for membership.

 Set builder notation:

B={x| x is a positive, even integer } defines set

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Basic sets of numbers

Symbol Set Example of members

Z Integers -3, 0, 2, 145 Q Rational numbers -5/6, 0, 1, 23/4 R Real numbers

Basic properties of sets

_{Sets are inherently unordered:}

_{No matter what objects a, b, and c denote,}

{a, b, c} = {a, c, b} = {b, a, c} = {b, c, a} = {c, a, b} = {c, b, a}.

_{All elements are distinct (unequal);} multiple listings make no difference!

{a, b, c} = {a, a, b, a, b, c, c, c, c}.

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Membership

_{The basic relationship between set and}

element is the inclusion membership. We write xS when element x is in set S and

xS otherwise. For instance

3{x| x is a positive, odd integer}

Definition of Set Equality

 Two sets are declared to be equal if and only if

they contain exactly the same elements.

 In particular, it does not matter how the set is defined or denoted.

 For example: The set {1, 2, 3, 4} =

{x | x is an integer where x>0 and x<5 } = {x | x is a positive integer whose square is >0 and <25}

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The Empty Set

 _{(“null”, “the empty set”) is the unique set} that contains no elements whatsoever.

_{The proposition}

x  is false for any element x.

x  is true for any element x.

_{We should distinguish between sets}  _and

{}. the first is the empty set, and the second is the set with exactly one element equal to empty set. So proposition {} is true.

Subset and Superset Relations

_S_{T (“S is a subset of T”) means that every} element of S is also an element of T.

_Examples:

{2,4,6} {x| x is positive, even integer}

S the empty set is subset of any set.

SS each set is subset of itself.

_S_{T (“S is a superset of T”) means T}_S.

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Venn diagrams for sets

_S_{T (“S is a proper subset of T”) means that}

ST but . Similar for ST.

S

T

Venn Diagram equivalent of ST

Example: {1,2}  {1,2,3}

S T _/

The

Power Set

Operation

_{The power set P(S) of a set S is the set of all} subsets of S. P(S) = {x | xS}.

_{E.g. P({a,b}) = {}_{, {a}, {b}, {a,b}}.} _{Sometimes P(S) is written 2}S.

Note that for finite S, |P(S)| = 2|S|.

_{It turns out that |P(N)| > |N|.}

Note: the above inequality is true also for infinite sets. There are different sizes of

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Ordered

n

-tuples

_{For n}_{N, an ordered n-tuple or a sequence}

of length n is written (a₁, a₂, …, a_n). The

first element is a₁, etc.

_{These are like sets, except that duplicates} matter, and the order makes a difference. _{Note (1, 2)}  _{(2, 1)}  _{(2, 1, 1).}

The Union Operator

_{For sets A, B, their union A}_{B is the set}

containing all elements that are either in A, or

in B (or, of course, in both).

_{Note that AB contains all the elements of A} and it contains all the elements of B: so it is true (AB  A) and (AB  B).

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The Union Operator

_{It is possible to represent inclusions (A}_B 

A) and (AB  B) using logical rules:

(xA (xA  xB ) (xB (xA  xB )

Both are special cases of tautology

_{a,b,c}_{{2,3} = {a,b,c,2,3}}

_{2,3,5}_{{3,5,7} = {2,3,5,3,5,7} ={2,3,5,7}} Venn diagram of the union of sets.

Union Examples

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The Intersection Operator

For sets A, B, their intersection AB is the set containing all elements that are simultaneously in A and in B.

xAB  (xA  xB ) AB A

AB B

based on tautology P  QP

_{-2,-3}_{{2,3} =} _{, Sets are disjoint} _{2,4,6}_{{3,4,5} ={4}}

Intersection Examples

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3

5

6

4

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The Difference Operator

_{For sets A, B, their difference A-B is the set} containing all elements that are in A and are not in B.

_{Formally, A,B: A-B{x | xA  xB}.}

_{Note that A-B is a subset of A and it is not a} subset of B: (A-B  A) and (A-B  B)

_{{2,3}-{2,3,4,5} =}

_{{2,4,6}-{3,4,5} ={2,6}} Venn diagram:

Difference Examples

3

5

4

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The Complement Operator

_{Sometimes we are dealing with sets that are all} subsets of a set U. This set is called a

universal set or universe.

_{For sets {1,2,3} {-1,0,4} {x| x is positive}

integer} a universal set can be chosen as

Z={x| x is integer}.

_{Given a universal set U, the set U-X is called} the complement of X and written X.

Set Identities - 1

_{Identity: A}_{=A A}_U=A _{Domination: A}_{U=U A}₌ _{Idempotent: AA = A = AA} _{Double complement:}

_{Commutative: AB=BA AB=BA} _{Associative: A(BC)=(AB)C}

A(BC)=(AB)C

A

A

)



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Set Identities - 2

_{Distributive: A}_(B_C)=(A_B)_(A_C)

A(BC)=(AB) (AC) _{Absorption: A}_(A_B)=A

A(AB)=A

_{Complement: A}_{A =U}

A A = 

_{De Morgan’s: (A}_{B)= A}_B

Generalized Union

_{Binary union operator: A}_B _{n-ary union:}

AA₂…A_n : ((…((A₁ A₂) …) A_n) (grouping & order is irrelevant)

_{“Big U” notation:}

_{Or for infinite sets of sets:}



1 



A

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Generalized Intersection

_{Binary intersection operator: A}_B _{n-ary intersection:}

AA₂…A_n((…((A₁A₂)…)A_n) (grouping & order is irrelevant)

_{“Big Arch” notation:}

_{Or for infinite sets of sets:}



1 



A

Cartesian Products of Sets

_{For sets A, B, their Cartesian product}

AB : {(a, b) | aA and bB }, it means set of all 2-tuples or ordered pairs.

_{E.g. {a,b}{1,2} = {(a,1),(a,2),(b,1),(b,2)}} _{Note that for finite A, B, |AB|=|A||B|.}

_{Note that the Cartesian product is not}

commutative, in general AB BA. But

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Cartesian Products of Sets

The Cartesian product of sets X₁,X₂, …X_n

is the set of all n-tuples (a₁,a₂, …a_n) where

a₁X₁,a₂X₂, …a_nX_n.

_{Example X={1,2}, Y={a,b}, Z={1,2},}

XYZ={(1,a,1), (1,a,2), (1,b,1), (1,b,2),

(2,a,1), (2,a,2), (2,b,1), (2,b,2)}. |XYZ|= =| X||Y||Z|= 8

_{The Cartesian product is not associative} (XY)Z X(YZ) XYZ