Fuzzy Logic - Classical Set Theory

Fuzzy Logic – Classical Set Theory

Fuzzy Logic – Classical Set Idea

set is an unordered assortment of various parts. It may be written explicitly by itemizing its parts utilizing the set bracket. If the order of the weather is modified or any ingredient of a set is repeated, it doesn’t make any modifications within the set.

Instance

  • A set of all constructive integers.
  • A set of all of the planets within the photo voltaic system.
  • A set of all of the states in India.
  • A set of all of the lowercase letters of the alphabet.

Mathematical Illustration of a Set

Units may be represented in two methods −

Roster or Tabular Kind

On this kind, a set is represented by itemizing all the weather comprising it. The weather are enclosed inside braces and separated by commas.

Following are the examples of set in Roster or Tabular Kind −

  • Set of vowels in English alphabet, A = {a,e,i,o,u}
  • Set of strange numbers lower than 10, B = {1,3,5,7,9}

Set Builder Notation

On this kind, the set is outlined by specifying a property that parts of the set have in frequent. The set is described as A = {x:p(x)}

Instance 1 − The set {a,e,i,o,u} is written as

A = {x:x is a vowel in English alphabet}

Instance 2 − The set {1,3,5,7,9} is written as

B = {x:1 ≤ x < 10 and (xpercent2) ≠ 0}

If a component x is a member of any set S, it’s denoted by x∈S and if a component y is just not a member of set S, it’s denoted by y∉S.

Instance − If S = {1,1.2,1.7,2},1 ∈ S however 1.5 ∉ S

Cardinality of a Set

Cardinality of a set S, denoted by |S||S|, is the variety of parts of the set. The quantity can be referred because the cardinal quantity. If a set has an infinite variety of parts, its cardinality is ∞∞.

Instance − |{1,4,3,5}| = 4,|{1,2,3,4,5,…}| = ∞

If there are two units X and Y, |X| = |Y| denotes two units X and Y having identical cardinality. It happens when the variety of parts in X is precisely equal to the variety of parts in Y. On this case, there exists a bijective perform ‘f’ from X to Y.

|X| ≤ |Y| denotes that set X’s cardinality is lower than or equal to set Y’s cardinality. It happens when the variety of parts in X is lower than or equal to that of Y. Right here, there exists an injective perform ‘f’ from X to Y.

|X| < |Y| denotes that set X’s cardinality is lower than set Y’s cardinality. It happens when the variety of parts in X is lower than that of Y. Right here, the perform ‘f’ from X to Y is injective perform however not bijective.

If |X| ≤ |Y| and |X| ≤ |Y| then |X| = |Y|. The units X and Y are generally referred as equal units.

Kinds of Units

Units may be categorised into many sorts; a few of that are finite, infinite, subset, common, correct, singleton set, and so forth.

Finite Set

A set which incorporates a particular variety of parts is known as a finite set.

Instance − S = x ∈ N and 70 > x > 50

Infinite Set

A set which incorporates infinite variety of parts is known as an infinite set.

Instance − S = x ∈ N and x > 10

Subset

A set X is a subset of set Y (Written as X ⊆ Y) if each ingredient of X is a component of set Y.

Instance 1 − Let, X = {1,2,3,4,5,6} and Y = {1,2}. Right here set Y is a subset of set X as all the weather of set Y is in set X. Therefore, we will write Y⊆X.

Instance 2 − Let, X = {1,2,3} and Y = {1,2,3}. Right here set Y is a subset (not a correct subset) of set X as all the weather of set Y is in set X. Therefore, we will write Y⊆X.

Correct Subset

The time period “correct subset” may be outlined as “subset of however not equal to”. A Set X is a correct subset of set Y (Written as X ⊂ Y) if each ingredient of X is a component of set Y and |X| < |Y|.

Instance − Let, X = {1,2,3,4,5,6} and Y = {1,2}. Right here set Y ⊂ X, since all parts in Y are contained in X too and X has a minimum of one ingredient which is greater than set Y.

Common Set

It’s a assortment of all parts in a specific context or software. All of the units in that context or software are primarily subsets of this common set. Common units are represented as U.

Instance − We could outline U because the set of all animals on earth. On this case, a set of all mammals is a subset of U, a set of all fishes is a subset of U, a set of all bugs is a subset of U, and so forth.

Empty Set or Null Set

An empty set incorporates no parts. It’s denoted by Φ. Because the variety of parts in an empty set is finite, empty set is a finite set. The cardinality of empty set or null set is zero.

Instance – S = x ∈ N and seven < x < 8 = Φ

Singleton Set or Unit Set

A Singleton set or Unit set incorporates just one ingredient. A singleton set is denoted by {s}.

Instance − S = x ∈ N, 7 < x < 9 = {8}

Equal Set

If two units comprise the identical parts, they’re mentioned to be equal.

Instance − If A = {1,2,6} and B = {6,1,2}, they’re equal as each ingredient of set A is a component of set B and each ingredient of set B is a component of set A.

Equal Set

If the cardinalities of two units are identical, they’re known as equal units.

Instance − If A = {1,2,6} and B = {16,17,22}, they’re equal as cardinality of A is the same as the cardinality of B. i.e. |A| = |B| = 3

Overlapping Set

Two units which have a minimum of one frequent ingredient are known as overlapping units. In case of overlapping units −

n(AB)=n(A)+n(B)n(AB)n(A∪B)=n(A)+n(B)−n(A∩B)
n(AB)=n(AB)+n(BA)+n(AB)n(A∪B)=n(A−B)+n(B−A)+n(A∩B)
n(A)=n(AB)+n(AB)n(A)=n(A−B)+n(A∩B)
n(B)=n(BA)+n(AB)n(B)=n(B−A)+n(A∩B)

Instance − Let, A = {1,2,6} and B = {6,12,42}. There’s a frequent ingredient ‘6’, therefore these units are overlapping units.

Disjoint Set

Two units A and B are known as disjoint units if they don’t have even one ingredient in frequent. Subsequently, disjoint units have the next properties −

n(AB)=ϕn(A∩B)=ϕ
n(AB)=n(A)+n(B)n(A∪B)=n(A)+n(B)

Instance − Let, A = {1,2,6} and B = {7,9,14}, there may be not a single frequent ingredient, therefore these units are overlapping units.

Operations on Classical Units

Set Operations embrace Set Union, Set Intersection, Set Distinction, Complement of Set, and Cartesian Product.

Union

The union of units A and B (denoted by A ∪ BA ∪ B) is the set of parts that are in A, in B, or in each A and B. Therefore, A ∪ B = x ∈ A OR x ∈ B.

Instance − If A = {10,11,12,13} and B = {13,14,15}, then A ∪ B = {10,11,12,13,14,15} – The frequent ingredient happens solely as soon as.

Union Operation

Intersection

The intersection of units A and B (denoted by A ∩ B) is the set of parts that are in each A and B. Therefore, A ∩ B = x ∈ A AND x ∈ B.

Intersection Operation

Distinction/ Relative Complement

The set distinction of units A and B (denoted by A–B) is the set of parts that are solely in A however not in B. Therefore, A − B = x ∈ A AND x ∉ B.

Instance − If A = {10,11,12,13} and B = {13,14,15}, then (A − B) = {10,11,12} and (B − A) = {14,15}. Right here, we will see (A − B) ≠ (B − A)

Relative Complement Operation

Complement of a Set

The complement of a set A (denoted by A′) is the set of parts which aren’t in set A. Therefore, A′ = x ∉ A.

Extra particularly, A′ = (U−A) the place U is a common set which incorporates all objects.

Instance − If A = x belongs to set of add integers then A′ = y doesn’t belong to set of strange integers

Complement of Set

Cartesian Product / Cross Product

The Cartesian product of n variety of units A1,A2,…An denoted as A1 × A2…× An may be outlined as all potential ordered pairs (x1,x2,…xn) the place x1 ∈ A1,x2 ∈ A2,…xn ∈ An

Instance − If we take two units A = {a,b} and B = {1,2},

The Cartesian product of A and B is written as − A × B = {(a,1),(a,2),(b,1),(b,2)}

And, the Cartesian product of B and A is written as − B × A = {(1,a),(1,b),(2,a),(2,b)}

Properties of Classical Units

Properties on units play an necessary position for acquiring the answer. Following are the totally different properties of classical units −

Commutative Property

Having two units A and B, this property states −

AB=BAA∪B=B∪A
AB=BAA∩B=B∩A

Associative Property

Having three units AB and C, this property states −

A(BC)=(AB)CA∪(B∪C)=(A∪B)∪C
A(BC)=(AB)CA∩(B∩C)=(A∩B)∩C

Distributive Property

Having three units AB and C, this property states −

A(BC)=(AB)(AC)A∪(B∩C)=(A∪B)∩(A∪C)
A(BC)=(AB)(AC)A∩(B∪C)=(A∩B)∪(A∩C)

Idempotency Property

For any set A, this property states −

AA=AA∪A=A
AA=AA∩A=A

Id Property

For set A and common set X, this property states −

Aφ=AA∪φ=A
AX=AA∩X=A
Aφ=φA∩φ=φ
AX=XA∪X=X

Transitive Property

Having three units AB and C, the property states −

If ABCA⊆B⊆C, then ACA⊆C

Involution Property

For any set A, this property states −

A¯¯¯¯¯¯¯¯=AA¯¯=A

De Morgan’s Regulation

It’s a essential regulation and helps in proving tautologies and contradiction. This regulation states −

AB¯¯¯¯¯¯¯¯¯¯¯¯¯=A¯¯¯¯B¯¯¯¯A∩B¯=A¯∪B¯
AB¯¯¯¯¯¯¯¯¯¯¯¯¯=A¯¯¯¯B¯¯¯¯