Sets as Mathematical Objects

• This module introduces sets as mathematical objects.
• Understanding set properties and operations enhances the efficient use of mathematical elements.
• Essential knowledge of sets and properties aids in analyzing vast amounts of data.

Learning Outcomes

• Determine whether a collection constitutes a set.
• Apply set operations to solve survey-related problems.
• Establish the equivalence of two sets through 1-1 correspondence.
• Recognize varying degrees of infinities.

Sets as Mathematical Objects

• A set is a well-defined collection where it's possible to determine if any given object is either in or not in the collection.
• An object belonging to a set is an element of that set.
• Uppercase letters denote sets, e.g., A, B, C.
• x ∈ A means that an object x belongs to the set A
• x ∉ A means that x does not belong to A.
• Sets are described through enumeration by listing elements, or by describing the elements using a rule.

Examples of sets defined using the enumeration method examples include

• V = {a, e, i, o, u}
• E = {2,4,6, 8,...}
• These sets can also be described using the rule method.
• V = {x | x is a vowel in the English alphabet}
• E = {x | x is a positive even number}
• a ∈ V and 2 ∈ E, while b ∉ V and 3 ∉ E.
• Order is not important.
• {a, e, I, o, u} is the same as {e, a, o, u, i} or {u, o, I, e, a}.

Equality

• Two sets A and B are equal (A = B) if they contain exactly the same elements.
• Equality is a relation on sets
• x ∈ A if and only if x ∉ B.

Non-sets

• {x | x is a large number} and {x | x is beautiful} are not sets as membership is unclear.
• "Large" is subjective and can vary.

Terminologies and notations

• Sets are either finite or infinite.
• A finite set has a listable number of elements otherwise it is infinite.
• Set V is finite, set E is infinite.
• The cardinality of a set means the number of elements it contains.
• The cardinality of set V, n(V) = 5.
• The cardinality of set E is not finite.
• An empty or null set has no elements, denoted by {} or Ø.
• The universal set, U, includes all elements in a specific context.
• Sets can be represented using Venn diagrams.

Using Venn Diagram

• Sets A and B are contained in a universal set U.
• U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
• A = {2, 4, 6, 8},
• B = {3, 6, 9}
• U is a rectangle, while A and B are circles.

Operations on Sets and Their Properties

• Operations are defined assuming sets A and B reside in a universal set U.

Operations

• Union of A and B
• A ∪ B = { x | x ∈ A and x ∈ B}
• Intersection of A and B
• A ∩ B = { x | x ∈ A or x ∈ B}
• Set difference of A and B
• A \ B = { x | x ∈ A and x ∉ B}
• Complement of A
• A^C = { x | x ∈ U and x ≠ A}.
• Cartesian Product of A and B
• A x B = {(a, b) | a ∈ A and b ∈ B}
• Union, Intersection, and Set Difference, plus the Cartesian product are binary operations like conjunction, disjunction, conditional and biconditional
• Complementation is a unary operation like negation.

Set operations examples

• U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
• A = {2, 4, 6, 8}
• B = {3, 6, 9}
• A ∪ B = {2, 3, 4, 6, 8, 9 }
• A ∩ B = {6}
• A \ B = { 2, 4, 8 }
• B \ A = { 3, 9 }
• A^C = { 1, 3, 5, 7, 9}
• (A∪B)^C = {1, 5, 7}
• A x B = {(2, 3), (2, 6), (2, 9), (4, 3), (4, 6), (4, 9), (6, 3), (6, 6), (6, 9), (8, 3), (8, 6), (8, 9)}
• B x A = {(3,2), (3, 4), (3, 6), (3, 8), (6,2), (6, 4), (6, 6), (6, 8), (9,2), (9, 4), (9, 6), (9, 8)}
• A \ B and B \ A are not equal.
• Cartesian products A x B and B x A are not equal.
• A and B belong to a universal set U
• Properties include:
  • Commutative Property:
    • A∪B=B∪A
    • A ∩ B=B∩A
  • Associative Property:
    • (A ∪ B) ∪ C = A ∪ (B ∪ C)
    • (A ∩ B) ∩ C = A ∩ (B ∩ C)

Set difference

• Set difference does not satisfy commutativity and associativity.

Formulas for cardinality

• n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
• n(A^C) = n(U) – n(A)
• n(A \ B) = n(A) – n(A ∩ B)
• n(A x B) = n(A) · n(B)

Example cardinality

• U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
• A = {2, 4, 6, 8}
• B = {3, 6, 9}.
• n(A) = 4
• n(B) = 3
• n(A ∩ B) = 1.
• n(A∪B) = n(A) + n(B) – n(A ∩ B) = 4 + 3-1 = 6
• n(A^C) = n(U) – n(A) = 9 – 4 = 5
• n(A \ B) = n(A) – n(A ∩ B) = 4 − 1 = 3
• n(A x B) = n(A) · n(B) = 4 · 3 = 12

Relations on Sets and their Properties

• A and B are in a universal set U.
• Equality means A and B are equal (A = B) if they have the same set of elements.
• x ∈ A if and only if x ∈ B.
• If A and B are not equivalent, write A ≠ B.

Subset

• Denoted A⊆ B, A is a subset of B or A is contained in B.
• Only if every element of A is also an element of B, if x ∈ A then x ∈ B.
• If A isn't a subset of B which is written A ≠ B.
• If A ⊆ B and B has an element not in A says A is a proper subset of B
• Denoted by A ⊂ B.

Equivalence

• A is equivalent to B, denoted A ~ B, if A and B have the same number of elements.
• Denoted n(A) = n(B).
• If A = B, then A ~ B
• “=” and “~” satisfy reflexivity, symmetry, and transitivity

Properties satisfied by ⊆ and ?

• Reflexivity dictates that for any set A, A = A and A ~ A.
• Symmetry states if A = B, then B = A; if A ~ B then B ~ A.
• Transitivity says if A = B and B = C, then A = C; if A ~ B and B~ C, then A ~C.

Ex

• V = {a, e, i o, u}
• E = { 2, 4, 6, ....}.
• L = {x | x is a vowel in the word “abstemious”}.
• L = V and L ~ V.
• It is evident that L ≠ E
• In fact, L and E are disjoint set since they have no recurring elements.
• M = {x | x is a letter in the word “abstemious”}.
• V ⊆ M.
• V is properly contained in M or V ⊂ M, because V only contains the vowels.

Using Sets in Survey Problems

• In a community of 200 residents, 135 were exposed to chemical X, 85 to Y, and 40 to X and Y.

Persons exposed:

• Chemical X only (X but not Y);
• Chemical Y only;
• X or Y;
• Neither chemical.
• Determine the set being looked at.

Let

• U be the universal set of 200 residents
• X residents exposed to chemical X
• Y residents exposed to chemical Y.

Given

• n(U) = 200
• n(X) = 135
• n(Y) = 85
• n(X ∩ Y) = 40.
• Chemical X only is X\Y, then n(X \ Y) = n(X) – n(X ∩ Y) = 135 – 40 = 95.
• Chemical Y only is Y\X, then n(Y\X) = n(Y) – n(Y ∩ X) = 85 – 40 = 45.
• Chemical X or Y is X ∪ Y, then n(X ∪ Y) = n(X) + n(Y) – n(X ∩ Y) = 135 + 85 – 40 = 180.
• n(X ∪ Y) is not n(X) + n(Y) since this gives n(XY) = 220, since there are only 200 in the study
• People exposed to neither chemical is n((X ∪ Y)^C) = n(U) – n(X ∪ Y) = 200 – 180 = 20.
• Use a Venn Diagram to answer questions.
• Cardinality rules also used.
• There were 200 surveyed, with 135 exposed to chemical X, and 85 to Y, with 40 to both.
• X ∩ Y which has 40 elements.

Cardinalities of Infinite Sets

• One-to-One Correspondence
• Two sets are equivalent if they have that same number of elements: n(A) = n(B).
• Equivalence using one-to-one correspondence.

Definition

• Two sets A and B have 1-1 correspondence every element of A maps to an element of B and vice versa.

• A function f: A → B has a correspondence with set B such that every element a ∈ A is associated with exactly one element b ∈ B.

• The function f: A → B is a bijection (or f is a bijective function) if every element of B maps to exactly one element of A, if a, c ∈ A and f(a) = f(c) = b, then a = c.

• Sets A and B are in 1-1 correspondence, there exists a bijection from a set A onto B is a bijective function.

• V= {a, e, i, o u} and A = {1, 2, 3, 4, 5} are equivalent since N(V) = n(A) =5 with a one-to-one correspondence

• The 1-1 correspondence or has multiple results

                A  V
                1 ↔ a
                2 ↔ e
                3 ↔ i
                4 ↔ o
                5 ↔ u
  

Formally,

• A set A is finite with cardinality (A) = k if a 1-1 correspondence with subset K= {1, 2, 3, ... k} of the natural numbers N exists.
• For finite sets if two sets A and B are finite with A ⊂ B, n(A) < n(B)
• This is not true with infinite sets.

Cardinality of the Set of Natural Numbers

• The set of natural numbers N = {1, 2, 3, 4, .....} is infinite in number.
• We define an infinite cardinality for N
• We define the cardinality of N to be (aleph-null)
• (N) =
• Other sets with this cardinality exist.
• Is the only infinite cardinal number?
• Is + 1 ?א + סא ?2 + א ?1 + סא
• E = {x | x is an even natural number = {2, 4, 6, 8, 10,...}
• O = {x | x is an odd natural number = {1, 3, 5, 7, 11, ....}

Other Sets to consider

• Counting numbers C = {0, 1, 2, 3, ....} = N ∪ {0}

• Integers Z = {...-k, ... -2, -1, -1, 0, 1, 2, 3, ...k,...}

• Rational Numbers*{a/b | a, b ∈ Z and b ≠ 0}*. (ℚ)

• Count the cardinalities of these infinite sets.

• For finite sets A and B, if A C B, then *n(A) < (B), this is not the case for infinite sets.

• E is an infinite set and a proper subset of N.

• odd natural numbers don't exist in E.

• establishes a 1-1 correspondence exists between N and E:

                N 1  2  3  4  5  k                 
                E  2  4  6  8  10 2k
  

• Every element k ∈ N maps to the element 2k ∈ E E of the form 2k (k ∈ N) maps to a unique k ∈ N.

• The two sets are equal, therefore n(N) = n(E( = א.

• O is also equal to N with 1-1 correspondence

                N  1  2  3  4  5 k
                O  1  3  5  7  9 2k - 1
  

• n(N) = n(O( =.

• A set can equal a subset fo itself.

Definition

• An infinite set is equal to a proper subset of itself.
• C, Z, and Q, contain N.
• C and Z are equal to N, C = NU {0} and Z is N with 0 and-N = {-k | k ∈ N}
• The set of rational numbers
• (ℚ) seems to have a lot more elements than N and is also its subset.
• Establish a 1-1 correspondence between N and Q
• Consider Q+, the set of positive rational numbers

N 1 2 3 4 5 k 1 1/1 2/1 3/1 4/1 5/1 k/1 2 1/2 2/2 3/2 4/2 5/2 k/2 3 1/3 2/3 3/3 4/3 5/3 k/3 4 1/4 4/2 3/4 4/4 5/4 k/4 . . . . . . m 1/m 2/m 3/m 4/m 5/m k/m

Note

• These all contain the positive rational numbers.

N 12345 k Row 1 1/1 2/1 3/1 4/1 5/1 k/1 Col 1 1/1 1/2 1/3 1/4 1/5

• This fails as the row and column don't correspond N elements.
• In summary this pairs N with elements in the following manner
• n(Q) = n(N( = א.
• Since C = N∪ {0} and n(C) = n(N), we can say that + 1 =
• • k = for any natural number k.

Rules of cardinal numbers

• Note that N = E∪ O and E and O are disjoint sets
• n(E ∪ O) = n(E) + n(O).
• • = . -Numbers given by the above table has א אelements but this set has cardinality
• Sets, N, C, Z, Q_all have , cardinality א are equivalent
• all are infinitely countable.

Defintion

• Definition 2.3. A set is said to be countable if it is finite or has cardinality א

The Cardinality of the Set of Real Numbers

• R, the the of real numbers, is the the union of *(ℚ)*rational numbers and (ℚ’) irrational numbers
• ∈ Q, it will either be expressed as a terminating decimal, rational numbers, or nonterminating, repeating decimals.
• Irrational numbers don not nonterminating, they do not the decimals.
• The set of real numbers is the set of both -∞ and ∞.
• This means each number on point corresponds value on the infinite line.
• N and R Equivalent?
• Georg Cantor: set theory mathematician, provides with to demonstrate that the is number theory
• Set -n(R) < of which X x ∈ X x ∈ to 1

π and the circle of gold ratios

• Assume n(R) < א.
• R* has values of the form 0.X1 X2 X3.... ..., where Xk ∈ n -For example, [0.12345,0.23456,0345670.45678,0.56789]
• Diagonal is the following
• Let that X1 ≠ 1 (First Number)
• Let that X2 ≠ 2 (Second Number)
• X5 ≠ 9 (Diagonal).

0.X1 X2 X3. π and the circle of gold rations

• Cantor's theorem of gold ratios

• "“enumerate" that can" gold, however π"

• Xm ≠ m. digit.

• Xm ≠ the theorem, that there" Cantor.

• The Gold circle.

• The line- Gold"

• To be x=-(-0,1),

• f(x) =Gold, that=((-π, π) and the value (golden=X+ Gold

• So I=R the is "Golden", I=R

• Is that I of the is can.

• What is the Real set, the Golden Value is.

Learn about sets as mathematical objects, their properties, and operations. This module helps in understanding how sets can be enumerated and described using rules. Grasping set theory is essential for solving survey problems and recognizing degrees of infinities.