MTH 225 Lecture-Module #08:
Sets



now covering Sections 2.1–2.2 of our textbook
(the material can be covered in different orders...)

sets are theoretical bases for statements about collections/groups
e.g. databases, linked lists, subclasses in Java/C++/etc., ...

a  is a collection or group of  which can be any objects

may be depicted pictorially as (sub)regions e.g. as follows:

where (as with predicates) should have specification of  U
from which elements may be drawn
here  U  is  {a,b,c,d,e,f,g,h,...,z}
and set B  contains  a,b,c,d,e
and set V  contains  a,e,i,o,u
and set E  contains  x,y,z}

in writing, instead of saying "contains", specify a set's list of elements with braces  
e.g.  B  is  {a,b,c,d,e}
e.g.  V  is  {a,e,i,o,u}
e.g. (with another universe)  {Color.RED, Color,GREEN, Color.BLUE}
e.g. (with another universe)  {1, 2, 3, 4}

another way a set may be specified is via the following notation:
    { expression : conditions_on_variables_in_expression }
    "set of all values of expression for which condition is satisfied"
reading ":" here as "such that"
e.g.:
    { x : x is a positive integer less than 5 }
    same as 
another e.g.:
    {x : x is a positive prime integer}
    which is  {2, 3, 5, 7, 11, 13, 17, 19, ...}
    (hard to write out complete, eh? ;-)

another e.g.:
    { x + y : x is an odd prime and y is an odd prime }
            :=  {}
   

some people use "|" instead of ":" in these set specifications
e.g.  { x | x is a positive integer less than 5 }
e.g.  {x | x is a positive prime integer}


order, multiplicity of elements not counted
e.g.  {red,green,blue}  considered equivalent to  {green,blue,red}
and the following two sets are considered equivalent to each other:



generally denote sets using uppercase letters (or such):
S,  A,  B,  S1,  S2, ...
e.g.  B  could be  {1,2,3,4}

empty set "{ }" may be specially written as "∅"

a set-specification can yield  ∅
e.g.:  {x : x < 5 and x > 12}
e.g.?:  {x : x is a pizza-topping and x is an ice-cream flavor}  ?
e.g.:  {x : false}


to indicate something is an element of a set,
use "∈"
syntax: "element ∈ set"
e.g.:  green ∈ {red,green,blue}

negation of "∈" normally expressed by drawing slash through it: "∉"
    like with "=", "≠"
e.g.:  yellow ∉ {red,green,blue}
e.g.:  green ∉ ∅
e.g.:  5 ∉ ∅
Axiom:  ∀x∀S[x ∉ S  ↔  ¬(x ∈ S)]
Axiom:  ∀x[x ∉ ∅]


we use the following symbols to denote the main sets of numbers:
"N" := natural numbers  {0,1,2,3,...}  a.k.a. nonnegative integers
"Z" := all integers  {...,-2,-1,0,1,2,...}  positive,negative,zero
"R := real numbers
    e.g.  √3 ∈ R
    e.g.  √(-1) ∉ R
"Q" := rational numbers
    { a/b : a ∈ Z  and  b ∈ Z  and  b ≠ 0 }
    Quotients

no a priori restriction on elements of set
e.g. could have  {red, green, -5, √3}
a set can even contain other sets
e.g.  { N, -2, R, yellow, {red,green} }
-2 ∈ { N, -2, R, yellow, {red,green} }
{red,green} ∈ { N, -2, R, yellow, {red,green} }
?  green ?∈? { N, -2, R, yellow, {red,green} } ?
    votes: yes: 
                    no: 
  abstentions: 

thus you can see: membership checked simply by checking equality to each element: 
  -2 =?= N
  -2 =?= -2
      .
      .
      .

  green =?= N
  green =?= -2
  green =?= R
  green =?= yellow
  green =?= {red,green}
it's common to use the term  for a set whose elements are sets
e.g.  { {a,b}, {a}, {b}, ∅ }  is a collection of sets
e.g.  { {a,b,c,d}, {e,f,g}, {h,i,j,k}, {l,m,n,o,p},
                {q,r,s}, {t,u,v}, {w,x,y,z} }
        is a collection of sets comprising all the letters
        in our alphabet
e.g. {N, Z, Q, R } is the collection of those main sets of numbers
e.g. { {1}, {1,2}, {1,3}, {1,2,4}, {1,5}, {1,2,3,6}, ... } is the collection of sets of positive integers' divisors

incidentally don't assume that  ∅  may be elided when it's a member of another set
e.g.  { {a,b}, {a}, {b}, ∅ }  not the same as  { {a,b}, {a}, {b} }
for an analogy, consider that  Strings are groups of characters somewhat like sets
e.g.:  "helo",  "world"
and  ∅  is like  ""  which is a valid  String
    can apply methods  .length(),  .substring(),  .toCharArray(),
    .compareTo(), etc. to  ""
    (may not get much from those method-calls for  "",
    but it is valid to do them)
then the collection {"helo", "world", ""}  contains how many  Strings? 
whereas by contrast the collection {"helo", "world"}  contains 


further operations on sets

well a set is like a  String  or array containing material
what can do with array besides accessing element?

  • array.length
    for a set, the notation for such determination of size is "|set|"
    called "cardinality"
    e.g.  |{red,green,blue}|  is 
    e.g.  |∅|  is 
    e.g.  |N|  is ∞
        rem.  {0,1,2,3,...}
    e.g.  |R|  is ∞
        (
        technically  |R|  different "∞" from  |N|
        explanation of that is beyond the scope of this course
        but it's presented at the end of our textbook's Section 2.4
        and if you want I could discuss with any of you outside of class
        e.g. during office-hours
        )
    e.g.  |{N}|  is 
        outer set has  inside
        { {0,1,2,3,...} }
    e.g.  |{∅}|  is 
        outer set has  inside
        { {} }
    watch braces as in code

  • "=": two sets equal iff contain same elements
    e.g.:
        {1,2,3,4}  =  {x : x is a positive integer less than 5}
        {red,green,blue}  =  {red,blue,green}
        {red,green,yellow}  ≠  {red,green,blue}
    Axiom:  ∀S1∀S2[S1 = S2  ↔  (∀x)(x ∈ S1  ↔  x ∈ S2)]

    is  {{red,green},yellow} ?=?/?≠? {red,green,yellow} ?


  • "⊂": containment
    syntax: "set1 ⊂ set2"
    semantics (i.e. meaning):  all the elements of set1 are also contained in set2
    we say "set1 is a subset of set2"
    and "set2 is a superset of set1"
    e.g.:  {green,blue} ⊂ {red,green,blue}
    e.g.:  {blue,green} ⊂ {red,green,blue}
    e.g.:  ¬({green,yellow} ⊂ {red,green,blue})
    e.g.?  {green} ?⊂? {red,green,blue} ? 
    e.g.?  green ?⊂? {red,green,blue} ?
    e.g.?  ∅ ?⊂? {red,green,blue} ?
        consider  {green,blue} ⊂ {red,green,blue}
        consider  {green} ⊂ {red,green,blue}
        consider  ∅ ⊂ {red,green,blue}
    e.g.?  {red} ?⊂? {red,{green,yellow}} ? 
    e.g.?  {green,yellow} ?⊂? {red,{green,yellow}} ? 
    e.g.:  {{green,yellow}} ⊂ {red,{green,yellow}}  is true
    e.g.:?  {red,green,blue} ?⊂? {red,green,blue} ? 

    we actually do say  {red,green,blue}  is a subset of itself,
    but just like with "<" and "≤",
    if it matters to you to disallow equality or allow it,
    with subset relationship use "⊂" or "⊆" respectively:
    e.g.:?  {red,green,blue} ?⊆? {red,green,blue} ? 
    e.g.:?  {red,green,blue} ?⊂? {red,green,blue} ? 
    e.g.:?  {green,blue} ?⊆? {red,green,blue} ? 
    e.g.?  ∅ ⊆ ∅ ? 
    e.g.?  ∅ ⊂ ∅ ? 
    in text, may add adjective "proper" for subset that is different from 'original' (⊂)
    e.g.: "{red,blue}  is a proper subset of  {red,green,blue}."
    e.g.: "{red,green,blue}  is not a proper subset of  {red,green,blue}."
    in fact all subsets of a set S are proper except S itself
    e.g. all the subsets — proper and itself — of  {a,b}  are as follows:
       
    thus,  { X : X ⊆ {a,b} }  i.e. { X : X is a subset of {a,b} }   is   { ∅, {a}, {b}, {a,b} }

    Axiom:  ∀S1∀S2[S1 ⊆ S2  ↔  ∀x(if  x ∈ S1  then  x ∈ S2)]
    Axiom:  ∀S1∀S2[S1 ⊂ S2  ↔  (S1 ⊆ S2  ∧  S1 ≠ S2)]

    if not subset, write something like "⊈" (a.k.a. "⊄" )
    e.g.  {1,4} ⊄ {1,2,3}


    of a set S is the collection of all the subsets of S
    denoted  ℘(S)
    e.g.  ℘({a,b}) = { ∅, {a}, {b}, {a,b} }
    e.g.  ℘({a}) = { ∅, {a} }
    e.g.  ℘(∅) = { ∅ }
    note  ℘(∅) ≠ ∅
    e.g.  ℘({a,b,c}) = { ∅, {a}, {b}, {a,b}, {a,c}, {b,c}, {a,c,b}, {c} }
    e.g.
    ℘({a,b,c,d}) = {
      ∅,  {a},   {b},   {a,b},   {a,c},   {b,c},   {a,c,b},   {c},
      {d}, {a,d}, {b,d}, {a,b,d}, {a,c,d}, {b,c,d}, {a,c,b,d}, {c,d}
      }

    observe: 
    |S|  |℘(S)|
-----------
 0     
 1     
 2     
 3     
 4     
 n

    rem. order,multiplicity of elements not significant for sets
    e.g.  {red,blue,green,red} = {red,green,blue}
    but order,multiplicity do matter for other data-structures
    such as (x,y)-coordinates
    e.g.:  (20,50) ?==? (50,20) ? 
    e.g.:  (20,20) ?==? (20) ? 
    (but  {20,20} = {20})

    Definition:
    a  (as in "quadruplet", "quintuplet", …) is a list of elements for which order does matter
    written with parentheses
        "(elt1, elt2, elt3, ..., eltn)"
    (naturally) a tuple with n elements called an "n-tuple"
    have more specific names for n-tuples with specific (low) values of n
    e.g. , , ...

    two tuples are equal iff they have equal length
    and elements at corresponding positions are equal
    e.g.:  (red,green,blue) == (red,green,blue)
    e.g.:  (red,blue,green) ≠ (red,green,blue)
    e.g.:  (20,20) ≠ (20)
    e.g.:  (20,50) ≠ (50,20)

    the following operation on sets involves tuples:

    Definition:
    If S1, S2, S3, ..., Sn are sets, then their ,
    denoted "S1 × S2 × ··· × Sn", is as follows:
        { every combination (a1,a2,...,an) : each ai ∈ Si }
    e.g.  {163} × {Dulimarta,Ferguson} × {easy,moderate,hard}

    e.g.  {t,f} × {t,f} := 
    e.g.  {t,f} × ∅ := { (t,), (f,) } ?
                                                        or maybe  { (t), (f) } ?
    consider
        {t,f} × {r,g,b}  :=
           

        {t,f} × {r,g}  :=
           

        {t,f} × {r}  :=
           

        {t,f} × {}  :=
           
    i.e.:
        {t,f} × ∅  :=
           


    again, may use  diagram
    to depict sets as regions of universe  U  of elements
    e.g. suppose  U  is  {a,b,c,d,e,f,g,h,...,z}
    and  B  is  {a,b,c,d,e}
    and  V  is  {a,e,i,o,u}
    and  E  is  {x,y,z}



    Definition:
    the  of sets  S1  and  S2, denoted "S1 ∪ S2",
    is the set
    containing all of  S1's elements plus all of  S2's elements listed together
    Axiom:   ∀S1∀S2∀x[x ∈ S1 ∪ S2  ↔  (x ∈ S1  or  x ∈ S2)]
    e.g.:  B ∪ V := 
    note  B ∪ V ≠ {{a,b,c,d,e},{a,e,i,o,u}}
    e.g.:  {a} ∪ {b,c} := {a,b,c}
    e.g.:   ∅ ∪ {a,b,c} := {a,b,c}

    another operation on pairs of sets is as follows:
    Definition: The  of sets S1 and S2, denoted "S1 ∩ S2",
        (a mnemonic for this symbol "∩" is "n" for "intersection")
    is the set comprising all the objects shared in common by S1 and S2
    Axiom:   ∀S1∀S2∀x[x ∈ S1 ∩ S2  iff  (x ∈ S1  and  x ∈ S2)]
    e.g.:  B ∩ V  is 
    another e.g.:   ∩   :=  ∅

    incidentally,   ∩ ∅ := ∅
    and   ∩ ∅ := ∅
    Theorem:   (∀S)[S ∩ ∅ := ∅]


    some further properties of "∪" and "∩" are as follows:

    Theorem:  (∀S1)(∀S2)(∀S3)[S1 ∩ (S2 ∪ S3)  =  (S1 ∩ S2) ∪ (S1 ∩ S3)]
        // distributivity of  ∩  over  ∪
    let  W := V ∪ {y} := 
    e.g. check
        W ∩ (B ∪ E) =?= (W ∩ B) ∪ (W ∩ E) ?
        W ∩  =?=  ∪  ?
         =?=  ?  

    Theorem:  (∀S1)(∀S2)(∀S3)[S1 ∪ (S2 ∩ S3)  =  (S1 ∪ S2) ∩ (S1 ∪ S3)]
        // distributivity of  ∪  over  ∩
    e.g. check
        B ∪ (W ∩ E) =?= (B ∪ W) ∩ (B ∪ E) ?
        B ∪  =?=  ∩  ?
         =?=  ?  

    Please use this sample material as models for your work for Exercise H.


    Definition:
    a pair of sets is called  if their intersection is empty.
    Axiom:   ∀S1∀S2[disjoint(S1,S2)  ↔  S1 ∩ S2 := ∅]
    a collection of sets {S1,S2,...,Sn} is called  if each pair Si,Sj of sets that are distinct (i.e. with i ≠ j) from the collection is disjoint
    e.g.  { {a,b,c,d}, {e,f,g}, {h,i,j,k}, {l,m,n,o,p},
            {q,r,s}, {t,u,v}, {w,x,y,z} }
    e.g.?  {B, V, E}  ? 
    e.g.?  {{r,g,b}, V, E}  ? 
    e.g.?  { {1}, {1,2}, {1,3}, {1,2,4}, {1,5}, {1,2,3,6}, ... }  ? 


    to  a set means to separate it into a collection of its subsets
    which are supposed to be 
    e.g.  { {1,2,3}, {4,5}, {6} }  is a partition of {1,2,3,4,5,6}
    e.g.  { {1,2,3,6}, {4,5} }  is a partition of {1,2,3,4,5,6}
    e.g.  { {1,2,4}, {3,5}, {4,6} }  doesn't qualify as a partition of {1,2,3,4,5,6}
    e.g.  { {1,2,4}, {3,6} }  doesn't qualify as a partition of {1,2,3,4,5,6}
    e.g. integers may be partitioned into negatives, zero, positives:
        { {...,-3,-2,-1}, {0}, {1,2,3,...} }
    e.g. U may be partitioned into  { B, M, E } 
            if M is 
    suppose set C comprises the consonants:
           
    e.g. ? then does  {C, V}  partition U ?
    partitioning really does happen in CS
    e.g. in compilers such as Javac/BlueJ/gcc
    at one stage in building parser (lexer),
    minimize a multi-state system by partitioning overall 'states'
    into sets of states


    further notation used for union, intersection of many sets
    e.g. suppose
    S1 = B = {a,b,c,d,e},
    S2 = V = {a,e,i,o,u},
    S3 = E = {x,y,z},
    S4 = {h,i,j,k},
    S5 = {v,w,x,y,z},
    S6 = something,
    S7 = something else,
    and so on up to S20,
    then instead of writing S1 ∪ S2 ∪ S3 ∪ S4 ∪ S5 ∪ S6 ∪ S7 ∪ S8 ∪ S9 ∪ S10 ∪ S11 ∪ S12 ∪ S13 ∪ S14 ∪ S15 ∪ S16 ∪ S17 ∪ S18 ∪ S19 ∪ S20,
    use notation as follows: 
     ∪ Si   // stands for "S1 ∪ S2 ∪ S3 ∪ . . . ∪ S20"
1≤i≤20
    and similarly for intersection: 
     ∩ Si   // stands for "S1 ∩ S2 ∩ S3 ∩ . . . ∩ S20"
1≤i≤20
    e.g. if S1 := {a,b,c,d}, S2 := {b,c,d,e}, and S3 := {c,d,e,f}, then: 
    ∪ Si  := 
1≤i≤3
    and: 
    ∩ Si  := 
1≤i≤3
    the expression under the "∪" or the "∩"
    is somewhat arbitrary
    e.g. for this example using numbers instead of letters,
        if Si is the set of positive divisors of i,
        S1:={1}, S2:={1,2}, S3:={1,3},
        S4:={1,2,4}, S5:={1,5}, S6:={1,2,3,6},
        and so on,
    then: 
    ∪Si  := 
 1≤i
    and: 
    ∩Si  := 
 1≤i
    in fact sometimes we write just the following: 
    ∪Si
    and: 
    ∩Si
    (maybe with an "i" under there)



    Definition:
    , denoted "S1 - S2" or maybe "S1 \ S2",
    yields all of  S1's elements that are not in  S2
    e.g.  B - V  :=  {b,c,d}
    Axiom:   ∀S1∀S2∀x[x ∈ S1 - S2  iff  (x ∈ S1  and  x ∉ S2)]

    incidentally, if  S1 ⊆ S2, how does  S1 - S2  look?
    e.g.  T = {a,i}
              T - V := 

    Definition:
    A set  S's , denoted "~S" or "Sc" or "S"
        or maybe "S'" or "S′"
    comprises everything not in S
    e.g.  ~V  is 
    a.k.a. "Vc" or "V" or maybe "V'" or "V′"
    Axiom:   ∀S∀x[x ∈ ~S  iff  x ∉ S]

    incidentally,  ~V  doesn't include  red  or  3
    because limit consideration to U universe of discourse:


    another e.g.  ~B := 
    a.k.a. "Bc" or "B" or maybe "B'" or "B′"

    incidentally  ~(S1 ∪ S2) == ~S1 ∩ ~S2
    e.g. considering the above two examples,
        ~B ∩ ~V := 
    and the elements not listed there are  {a,b,c,d,e,i,o,u}  which equals  B ∪ V
    thus  ~B ∩ ~V := ~(B ∪ V)

    set-properties such as that listed page  of our textbook
    can be proven from axioms
    e.g.  ∀S1∀S2[~(S1 ∪ S2) == ~S1 ∩ ~S2]

    another e.g.  (∀S)[~~S == S]
    e.g.  ~V  is everything not a vowel i.e. all the consonants,
    so  ~(~V)  is everything not a consonant i.e. all the vowels again, 



    to clarify things, just producing Venn diagram(s) is often useful
    e.g. Rosen 2.2.27a:  A ∩ (B - C)  (on board)
    e.g. Rosen 2.2.27c:  (A ∩ ~B) ∪ (A ∩ ~C)  (on board)


    sample proof:
    Proposition:  (∀S)[S ∪ ∅ = S]


    More concisely — without the explicative details:
    Proposition (same as above):  (∀S)[S ∪ ∅ = S]

    You can model your proofs based on this concise one.


    another sample proof:
    Proposition:  (∀S1)(∀S2)(S1 ∩ [S1 ∪ S2] = S1)

    You can model your proofs based on this one also.
    (see also our textbook's Exercise 2.2.13 and its solution in the back of the book.)

    (Copyright © 2009 by Hugh McGuire   — for thoughts about this, see   http://www.cis.gvsu.edu/~mcguire/teaching/copyright_thoughts.html .)