1. 陳光琦助理教授 (Kuang-Chi Chen) chichen6@mail.tcu.edu.tw
資訊科學數學5 : Sets Theory
陳光琦助理教授 (Kuang-Chi Chen)

2. Introduction to Set Theory
A set is a new type of structure, representing an unordered collection (group, plurality) of distinct (different) objects.
Set theory deals with operations between, relations among, and statements about sets.
Sets are ubiquitous in computer software systems.
All of mathematics can be defined in terms of some form of set theory.
Plurality 1.複數；多數；大多數；過半數

3. Basic Notations for Sets
For sets, we use capital letters 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 3 objects denoted by a, b, c.
A universe or universe of discourse which is usually denoted by U . We then select only elements from U to form our sets.
Read {a, b, c} as “the set whose elements are a, b, and c” or just “the set a, b, c”.
Universe 【邏輯學】整體 宇集

4. Basic Notations (cont’d)
Set builder notation: For any proposition P(x) over any universe of discourse, {x|P(x)} is “the set of all x such that P(x)”. Here, “ | ” is read “such that”.
E.g., if U denotes the set of all integers then {x | 1≤ x ≤5} would contain all of the integers between 1 and 5 inclusive, therefore it would be {1, 2, 3, 4, 5}.
Read {a, b, c} as “the set whose elements are a, b, and c” or just “the set a, b, c”.
Universe 【邏輯學】整體 宇集

5. 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 !
If a = b, then {a, b, c} = {a, c} = {b, c} = {a, a, b, a, b, c, c, c}.
This set contains (at most) 2 elements!

6. Definition of Set Equality
Two sets are declared to be equal iff they contain exactly the same elements.
In particular, it does not matter how the set is defined or denoted.
E.g., 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}

7. Infinite Sets
Conceptually, sets may be infinite (i.e., not finite, without end, unending).
Symbols for some special infinite sets: N = {0, 1, 2, …} The Natural numbers. Z = {…, -2, -1, 0, 1, 2, …} The Integers. R = The “Real” numbers, such as …
Blackboard Bold or double-struck font (ℕ, ℤ , ℝ) is also often used for the special number sets.
Infinite sets come in different sizes! (More on “functions”)

8. Venn Diagrams 2 4 6 8 3 1 -1 5 7 9 John Venn 1834-1923 Primes < 10-1 1 2 3 4 5 6 7 8 9
Even integers from 2 to 9
With Venn diagrams, you can see that one set is a subset of another just by seeing that you can draw an enclosure around its members that fits completely inside an enclosure drawn around the larger set’s members.
Integers from -1 to 9
Positive integers less than 10

9. Basic Set Relations

10. Basic Set Relations
xS (“x is in S”) is the proposition that object x is an element or member of set S.
E.g., 3N, “a”{x | x is a letter of the alphabet}
xS : (xS) “x is not in S”
alphabet 1.字母表。

11. The Empty Set
 (“null”, “empty set”) is the unique set that contains no elements whatsoever.
 = { } = {x | False}
No matter the domain of discourse, we have the axiom x: x.

12. Subset and Superset Relations
ST (“S is a subset of T”) means that “every element of S is an element of T ”.
ST  x (xS  xT)
S, SS.
ST (“S is a superset of T”) means TS.
means (ST), i.e. x (xS  xT).
S 包含於 T (S is a subset of T), i.e., T 包含 S (T is a superset of S)
Note also that FORALL x P(x)->Q(x) can also be understood as meaning “{x|P(x)} is a subset of {x|Q{x}}”. This can help you understand the meaning of implication. For example, if I say, “if a student has a drivers license, then he is over 16,” this is the same as saying “the set of students with drivers licenses is a subset of the set of students who are over 16”, or “every student with a drivers license is over 16.” If no students in the universe of discourse have drivers licenses, then the antecedent is always false, or in other words the set of students with drivers licenses is just the empty set, which is of course a member of every set, and so the statement is vacuously true. Alternatively, if every student in the universe of discourse is over 16, then the consequent is always true, that is, the set of students who are over 16 is the entire universe of discourse, and so every set of students in the u.d. is necessarily a subset of the set of students who are over 16, and so the statement is trivially true. The statement is only false if there exists a student with a drivers license in the u.d. who is under 16 (perhaps the license is fake or from a foreign country), in which case, the set of students with drivers licenses is *not* a subset of the under-16 students.

13. Proper (Strict) Subsets
ST (“S is a proper subset of T”) means that ST but S≠T , i.e.,
“ T contains some elements that are not in S ”. S T
Venn Diagram equivalent of ST
Example: {1, 2}  {1, 2, 3}
We may also say, “S is a strict subset of T”, or “S is strictly a subset of T” to mean the same thing.

14. About Subset & Proper Subset
ST ⇒ ST .
If S, T are finite, ST ⇒|S| ≤ |T| , and
ST ⇒|S| < |T| ,
where “|S|” denotes the number of elements in S and is referred to as the cardinality or size of S.
However, we do not find that ST ⇒ ST .
Cardinality 3.〔常 pl. 〕基數。

15. Set Equality Define set equality in terms of  relation:
S, T : S = T  (x : xS  xT)
“ Two sets are equal iff
they have all the same members ”
Note that S=T  ST ST.

16. Sets Are Objects
The objects that are elements of a set may themselves be sets.
E.g., let S={x | x  {1,2,3}}
then S={ , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }
Note that 1  {1}  {{1}} !!!
In general, any kind of object or structure, whether simple or complex, can be a member of a set. In particular, sets themselves (being structures) can be members of sets.

17. Cardinality and Finiteness
|S| (read “the cardinality of S”) is a measure of how many different elements S has.
E.g., ||=0, |{1, 2, 3}| = 3, |{a, b}| = 2, |{{1,2,3}, {4,5}}| = __
What are some infinite sets we’ve seen?
N, Z, R 2
Cardinality 3.〔常 pl. 〕基數。
If |S|N, then we say S is finite.??? Otherwise, we say S is infinite. ……..No! no! no! ???

18. More Finiteness
E.g., for U = {1, 2, 3, 4, …}, the set of positive integers, let
a) A = {1, 4, 9, …, 64, 81} = {x2| xU , x2 < 100} = {x2| xU  x2 < 100} .
b) B = {1, 4, 9, 16} = {y2| yU , y2 < 20} = {y2| yU  y2 < 17} .
c) C = {2, 4, 6, 8, …} = {2k| kU}
Sets A and B are finite sets, whereas C is an infinite set.

19. The Power Set Operation
The power set of a set S, P(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 2S. Note that for finite S, |P(S)| = 2|S|.
It turns out S: |P(S)|>|S| ,
e.g., |P(N)| > |N|.
There are different sizes of infinite sets!
Power冪
We’ll get to different sizes of infinite sets later, in the module on functions.

20. Set Notations …so far… Variable objects x, y, z; sets S, T, U.
Literal set {a, b, c} and set-builder {x|P(x)}.
 relational operator, and the empty set .
Set relations = ,  ,  ,  ,  ,  , etc.
Venn diagrams.
Cardinality |S| and infinite sets N, Z, R.
Power sets P(S).
Literal 用字母代表的

21. Naïve Set Theory is Inconsistent
There are some naïve set descriptions that lead to pathological structures that are not well-defined.
(That do not have self-consistent properties.)
These “sets” mathematically cannot exist.
E.g., let S = {x | xx }. Is SS?
Therefore, consistent set theories must restrict the language that can be used to describe sets.
Don’t worry about it in this class !
pathological病理學(上)的；病態的

22. Naïve Set Theory is Inconsistent
A well-defined set implies that for any element we care to consider, we are able to determine whether it is in the set under scrutiny. Consequently, we avoid dealing with sets that depend on opinion, such as the set of outstanding major league pitchers in 1980s.
(That do not have self-consistent properties.)
scrutiny 1.細看；細讀。2.仔細檢查[考察]；復查；徹查。
Bertrand Russell was the guy who discovered this particular pathological example. The question of whether it is a member of itself is known as “Russell’s Paradox.”
Bertrand Russell
Bertrand Russell was the guy who discovered this particular pathological example. The question of whether it is a member of itself is known as “Russell’s Paradox.”

23. Ordered n-tuples
These are like sets, except that duplicates matter, and the order makes a difference.
For nN, an ordered n-tuple or a sequence or list of length n is written (a1, a2, …, an). Its first element is a1, etc.
Note that (1, 2)  (2, 1)  (2, 1, 1).
Empty sequence, singlets, pairs, triples, quadruples, quintuples, …, n-tuples.
Contrast with sets { }
Quintuples, 1.五的；五倍的；五重的；由五個部份組成的。1.五倍量。使成五倍，以五乘之。
N-tuple .n倍量。
Sometimes people also define “bags”, which are unordered collections in which duplicates matter. If you have a bag of coins, they are in no particular order, but it matters how many coins of each type you have.

24. Cartesian Products of Sets
For sets A, B, their Cartesian product AB : {(a, b) | aA  bB }.
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: the Cartesian product is not commutative, i.e., A, B: AB=BA.
Extends to A1  A2  …  An ...
Cartesian (法國哲學家)笛卡兒(Descartes) 的
Usually AxBxC is defined as {(a,b,c) | a is in A and b is in B and c is in C}.
René Descartes ( )

25. Review Sets S, T, U…; Special sets N, Z, R.
Set notations {a, b,...}, {x|P(x)}…
Set relation operators xS, ST, ST, S=T, ST, ST. (These form propositions.)
Finite vs. infinite sets.
Set operations |S|, P(S), ST.
Next: more set operators: , , .
“|S|” denotes the number of elements in S and is referred to as the cardinality or size of S

26. Set Operators

27. 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).
Formally, A, B: AB = {x | xA  xB}.
Note that AB is a superset of both A and B (in fact, it is the smallest such superset):
A, B: (AB  A)  (AB  B)

28. Union Examples
The switches in a network need not act independently of each other.
E.g., {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}
Union 2 3 5 7

29. 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.
Formally, A,B: AB={x | xA  xB}.
Note that AB is a subset of both A and B (in fact it is the largest such subset):
A, B: (AB  A)  (AB  B)

30. Intersection Examples
{a,b,c}{2,3} = 
{2,4,6}{3,4,5} = {4} 2 3 5 6 4
Intersection

31. Disjointedness
Two sets A, B are called disjoint (i.e., unjoined) iff their intersection is empty. (AB=)
E.g., the set of even integers is disjoint with the set of odd integers. 2 6 4 … -2 -4 3 -1 1 5 9 7 -3 …
Disjointedness

32. Inclusion-Exclusion Principle
How many elements are in AB ? |AB| = |A|  |B|  |AB|
E.g., How many students are on our class list? Consider set E  I  M, I = {s | s turned in an information sheet} M = {s | s sent the TAs their address}
Some students did both! |E| = |IM| = |I|  |M|  |IM|
Subtract out items in intersection, to compensate for double-counting them.
We see this basic counting principle too when we talk about combinatorics.

33. Set Difference Examples
For sets A, B, the difference of A and B, written AB, is the set of all elements that are in A but not B. Formally:
A  B : x  xA  xB  x  xA  xB 
Also called: The complement of B with respect to (w.r.t.) A.
NOT (x in A -> x in B) = NOT (x not in A or x in B) (defn. of implies) = x in A AND x not in B (DeMorgan’s law).

34. Set Difference Examples
{1, 2, 3, 4, 5, 6}  {2, 3, 5, 7, 9, 11}
= {1, 4, 6}
Z  N  {… , −1, 0, 1, 2, … }  {0, 1, … } = {x | x is an integer but not a natural #} = {x | x is a negative integer} = {… , −3, −2, −1}

35. Set Difference - Venn Diagram
A−B is what’s left after B “takes a bite out of A”
Set A
Set B
Set AB

36. Set Complements
The universe of discourse can itself be considered a set, call it U.
When the context clearly defines U, we say that for any set A U, the complement of A, written , is the complement of A w.r.t. U, i.e., it is U  A.
E.g., If U = N, A={3, 5}, then
U  A = {0, 1, 2, 4, 6, 7, …}

37. More on Set Complements
An equivalent definition, when U is clear : A U
Note that set difference and complement do not relate to each other like arithmetic difference and negative. In arithmetic, we know that a-b = -(b-a). But in sets, A-B is not generally the same as the complement of B-A.

38. Set Identities Identity: A = A = AU 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

39. DeMorgan’s Law for Sets
Exactly analogous to (and provable from) DeMorgan’s Law for propositions.

40. Proving Set Identities
# To prove statements about sets, of the form E1 = E2 (where the E’s are set expressions), here are three useful techniques:
1. Prove E1  E2 and E2  E1 separately.
2. Use set builder notation & logical equivalences.
(before and after)
3. Use a membership table.
(A membership table is like a truth table)
A membership table is like a truth table.

41. Method 1: Mutual Subsets
E.g., Show A(BC) = (AB)(AC).
Part 1: Show A(BC)  (AB)(AC).
Assume xA(BC), show that x(AB)(AC).
We know that xA, and either xB or xC.
Case 1: xB, then xAB, so x(AB)(AC).
Case 2: xC, then xAC, so x(AB)(AC).
Therefore, x(AB)(AC).
Hence, A(BC)(AB)(AC).
Part 2: Show (AB)(AC)  A(BC).

42. Method 3: Membership Tables
Just like truth tables for propositional logic.
Columns for different set expressions.
Rows for all combinations of memberships in constituent sets.
Use “1” to indicate membership in the derived set, “0” for non-membership.
Prove equivalence with identical columns.

43. Membership Table Example
Prove (AB)B = AB.

44. Membership Table Exercise
Prove (AB)C = (AC)(BC).

45. Review Sets S, T, U… Special sets N, Z, R.
Set notations {a,b,...}, {x|P(x)}…
Relations xS, ST, ST, S=T, ST, ST.
Operations |S|, P(S), , , , ,
Set equality proof techniques:
Mutual subsets.
Derivation using logical equivalences.

46. Generalized Unions & Intersections
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 (A1, …, An), or even on unordered sets of sets, X={A | P(A)}.

47. Generalized Union Binary union operator: AB
n-ary union: A1A2…An : ((…((A1 A2) …) An) (grouping & order is irrelevant)
“Big U” notation:
Or for infinite sets of sets:

48. Generalized Intersection
Binary intersection operator: AB
n-ary intersection: A1A2…An((…((A1A2)…)An) (grouping & order is irrelevant)
“Big Arch” notation:
Or for infinite sets of sets:
N-tuple .n倍量。

49. Representations
A frequent theme of this course will be methods of representing one discrete structure using another discrete structure of a different type.
E.g., one can represent natural numbers as
Sets: 0:, 1:{0}, 2:{0, 1}, 3:{0, 1, 2}, …
Bit strings: 0:0, 1:1, 2:10, 3:11, 4:100, …

50. Representing Sets with Bit Strings
For an enumerable u.d. U with ordering x1, x2, …, represent a finite set SU as the finite bit string B = b1b2…bn where i: xiS  (i<n  bi=1).
E.g., U=N, S={2, 3, 5, 7, 11}, B=
In this representation, the set operators “”, “”, “” are implemented directly by bitwise OR, AND, NOT!
So, for example, on a 64-bit processor, using just a single machine-language instruction you can calculate the union or intersection of two sets out of a universe of discourse having up to 64 elements. This leads to an extremely fast way of doing complicated calculations on small sets. It is not a good method for large, sparsely populated sets, because searching the bit string to find which bits are “1” can take a long time.