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Inference System

 

Inference System

Fuzzy Inference System is the key unit of a fuzzy logic system having decision making as its primary work. It uses the “IF…THEN” rules along with connectors “OR” or “AND” for drawing essential decision rules.

Characteristics of Fuzzy Inference System

Following are some characteristics of FIS −

• The output from FIS is always a fuzzy set irrespective of its input which can be fuzzy or crisp.

• It is necessary to have fuzzy output when it is used as a controller.

• A defuzzification unit would be there with FIS to convert fuzzy variables into crisp variables.

Functional Blocks of FIS

The following five functional blocks will help you understand the construction of FIS −

• Rule Base − It contains fuzzy IF-THEN rules.

• Database − It defines the membership functions of fuzzy sets used in fuzzy rules.

• Decision-making Unit − It performs operation on rules.

• Fuzzification Interface Unit − It converts the crisp quantities into fuzzy quantities.

• Defuzzification Interface Unit − It converts the fuzzy quantities into crisp quantities. Following is a block diagram of fuzzy interference system.

Working of FIS

The working of the FIS consists of the following steps −

• A fuzzification unit supports the application of numerous fuzzification methods, and converts the crisp input into fuzzy input.

• A knowledge base - collection of rule base and database is formed upon the conversion of crisp input into fuzzy input.

• The defuzzification unit fuzzy input is finally converted into crisp output.

Methods of FIS

Let us now discuss the different methods of FIS. Following are the two important methods of FIS, having different consequent of fuzzy rules −

• Mamdani Fuzzy Inference System
• Takagi-Sugeno Fuzzy Model (TS Method)

Mamdani Fuzzy Inference System

This system was proposed in 1975 by Ebhasim Mamdani. Basically, it was anticipated to control a steam engine and boiler combination by synthesizing a set of fuzzy rules obtained from people working on the system.

Steps for Computing the Output

Following steps need to be followed to compute the output from this FIS −

• Step 1 − Set of fuzzy rules need to be determined in this step.

• Step 2 − In this step, by using input membership function, the input would be made fuzzy.

• Step 3 − Now establish the rule strength by combining the fuzzified inputs according to fuzzy rules.

• Step 4 − In this step, determine the consequent of rule by combining the rule strength and the output membership function.

• Step 5 − For getting output distribution combine all the consequents.

• Step 6 − Finally, a defuzzified output distribution is obtained.

Following is a block diagram of Mamdani Fuzzy Interface System.

Takagi-Sugeno Fuzzy Model (TS Method)

This model was proposed by Takagi, Sugeno and Kang in 1985. Format of this rule is given as −

IF x is A and y is B THEN Z = f(x,y)

Here, AB are fuzzy sets in antecedents and z = f(x,y) is a crisp function in the consequent.

Fuzzy Inference Process

The fuzzy inference process under Takagi-Sugeno Fuzzy Model (TS Method) works in the following way −

• Step 1: Fuzzifying the inputs − Here, the inputs of the system are made fuzzy.

• Step 2: Applying the fuzzy operator − In this step, the fuzzy operators must be applied to get the output.

Rule Format of the Sugeno Form

The rule format of Sugeno form is given by −

if 7 = x and 9 = y then output is z = ax+by+c

Comparison between the two methods

Let us now understand the comparison between the Mamdani System and the Sugeno Model.

• Output Membership Function − The main difference between them is on the basis of output membership function. The Sugeno output membership functions are either linear or constant.

• Aggregation and Defuzzification Procedure − The difference between them also lies in the consequence of fuzzy rules and due to the same their aggregation and defuzzification procedure also differs.

• Mathematical Rules − More mathematical rules exist for the Sugeno rule than the Mamdani rule.

• Adjustable Parameters − The Sugeno controller has more adjustable parameters than the Mamdani controller.

Approximate Reasoning

 

Approximate Reasoning

Following are the different modes of approximate reasoning −

Categorical Reasoning

In this mode of approximate reasoning, the antecedents, containing no fuzzy quantifiers and fuzzy probabilities, are assumed to be in canonical form.

Qualitative Reasoning

In this mode of approximate reasoning, the antecedents and consequents have fuzzy linguistic variables; the input-output relationship of a system is expressed as a collection of fuzzy IF-THEN rules. This reasoning is mainly used in control system analysis.

Syllogistic Reasoning

In this mode of approximation reasoning, antecedents with fuzzy quantifiers are related to inference rules. This is expressed as −

x = S1A′s are B′s

y = S2C′s are D′s

------------------------

z = S3E′s are F′s

Here A,B,C,D,E,F are fuzzy predicates.

• S1 and S2 are given fuzzy quantifiers.

• S3 is the fuzzy quantifier which has to be decided.

Dispositional Reasoning

In this mode of approximation reasoning, the antecedents are dispositions that may contain the fuzzy quantifier “usually”. The quantifier Usually links together the dispositional and syllogistic reasoning; hence it pays an important role.

For example, the projection rule of inference in dispositional reasoning can be given as follows −

usually( (L,M) is R ) ⇒ usually (L is [R ↓ L])

Here [R ↓ L] is the projection of fuzzy relation R on L

Fuzzy Logic Rule Base

It is a known fact that a human being is always comfortable making conversations in natural language. The representation of human knowledge can be done with the help of following natural language expression −

IF antecedent THEN consequent

The expression as stated above is referred to as the Fuzzy IF-THEN rule base.

Canonical Form

Following is the canonical form of Fuzzy Logic Rule Base −

Rule 1 − If condition C1, then restriction R1

Rule 2 − If condition C1, then restriction R2

.

.

.

Rule n − If condition C1, then restriction Rn

Interpretations of Fuzzy IF-THEN Rules

Fuzzy IF-THEN Rules can be interpreted in the following four forms −

Assignment Statements

These kinds of statements use “=” (equal to sign) for the purpose of assignment. They are of the following form −

a = hello

climate = summer

Conditional Statements

These kinds of statements use the “IF-THEN” rule base form for the purpose of condition. They are of the following form −

IF temperature is high THEN Climate is hot

IF food is fresh THEN eat.

Unconditional Statements

They are of the following form −

GOTO 10

turn the Fan off

Linguistic Variable

We have studied that fuzzy logic uses linguistic variables which are the words or sentences in a natural language. For example, if we say temperature, it is a linguistic variable; the values of which are very hot or cold, slightly hot or cold, very warm, slightly warm, etc. The words very, slightly are the linguistic hedges.

Characterization of Linguistic Variable

Following four terms characterize the linguistic variable −

• Name of the variable, generally represented by x.
• Term set of the variable, generally represented by t(x).
• Syntactic rules for generating the values of the variable x.
• Semantic rules for linking every value of x and its significance.

Propositions in Fuzzy Logic

As we know that propositions are sentences expressed in any language which are generally expressed in the following canonical form −

s as P

Here, s is the Subject and P is Predicate.

For example, “Delhi is the capital of India”, this is a proposition where “Delhi” is the subject and “is the capital of India” is the predicate which shows the property of subject.

We know that logic is the basis of reasoning and fuzzy logic extends the capability of reasoning by using fuzzy predicates, fuzzy-predicate modifiers, fuzzy quantifiers and fuzzy qualifiers in fuzzy propositions which creates the difference from classical logic.

Propositions in fuzzy logic include the following −

Fuzzy Predicate

Almost every predicate in natural language is fuzzy in nature hence, fuzzy logic has the predicates like tall, short, warm, hot, fast, etc.

Fuzzy-predicate Modifiers

We discussed linguistic hedges above; we also have many fuzzy-predicate modifiers which act as hedges. They are very essential for producing the values of a linguistic variable. For example, the words very, slightly are modifiers and the propositions can be like “water is slightly hot.”

Fuzzy Quantifiers

It can be defined as a fuzzy number which gives a vague classification of the cardinality of one or more fuzzy or non-fuzzy sets. It can be used to influence probability within fuzzy logic. For example, the words many, most, frequently are used as fuzzy quantifiers and the propositions can be like “most people are allergic to it.”

Fuzzy Qualifiers

Let us now understand Fuzzy Qualifiers. A Fuzzy Qualifier is also a proposition of Fuzzy Logic. Fuzzy qualification has the following forms −

Fuzzy Qualification Based on Truth

It claims the degree of truth of a fuzzy proposition.

Expression − It is expressed as x is t. Here, t is a fuzzy truth value.

Example − (Car is black) is NOT VERY True.

Fuzzy Qualification Based on Probability

It claims the probability, either numerical or an interval, of fuzzy proposition.

Expression − It is expressed as x is λ. Here, λ is a fuzzy probability.

Example − (Car is black) is Likely.

Fuzzy Qualification Based on Possibility

It claims the possibility of fuzzy proposition.

Expression − It is expressed as x is π. Here, π is a fuzzy possibility.

Example − (Car is black) is Almost Impossible.

Traditional Fuzzy Refresher

 

Traditional Fuzzy Refresher

Logic, which was originally just the study of what distinguishes sound argument from unsound argument, has now developed into a powerful and rigorous system whereby true statements can be discovered, given other statements that are already known to be true.

Predicate Logic

This logic deals with predicates, which are propositions containing variables.

A predicate is an expression of one or more variables defined on some specific domain. A predicate with variables can be made a proposition by either assigning a value to the variable or by quantifying the variable.

Following are a few examples of predicates −

• Let E(x, y) denote "x = y"
• Let X(a, b, c) denote "a + b + c = 0"
• Let M(x, y) denote "x is married to y"

Propositional Logic

A proposition is a collection of declarative statements that have either a truth value "true” or a truth value "false". A propositional consists of propositional variables and connectives. The propositional variables are dented by capital letters (A, B, etc). The connectives connect the propositional variables.

A few examples of Propositions are given below −

• "Man is Mortal", it returns truth value “TRUE”
• "12 + 9 = 3 – 2", it returns truth value “FALSE”

The following is not a Proposition −

• "A is less than 2" − It is because unless we give a specific value of A, we cannot say whether the statement is true or false.

Connectives

In propositional logic, we use the following five connectives −

• OR (∨∨)
• AND (∧∧)
• Negation/ NOT (¬¬)
• Implication / if-then (→→)
• If and only if (⇔⇔)

OR (∨∨)

The OR operation of two propositions A and B (written as A∨BA∨B) is true if at least any of the propositional variable A or B is true.

The truth table is as follows −

A	B	A ∨ B
True	True	True
True	False	True
False	True	True
False	False	False

AND (∧∧)

The AND operation of two propositions A and B (written as A∧BA∧B) is true if both the propositional variable A and B is true.

The truth table is as follows −

A	B	A ∧ B
True	True	True
True	False	False
False	True	False
False	False	False

Negation (¬¬)

The negation of a proposition A (written as ¬A¬A) is false when A is true and is true when A is false.

The truth table is as follows −

A	¬A
True	False
False	True

Implication / if-then (→→)

An implication A→BA→B is the proposition “if A, then B”. It is false if A is true and B is false. The rest cases are true.

The truth table is as follows −

A	B	A→B
True	True	True
True	False	False
False	True	True
False	False	True

If and only if (⇔⇔)

A⇔BA⇔B is a bi-conditional logical connective which is true when p and q are same, i.e., both are false or both are true.

The truth table is as follows −

A	B	A⇔B
True	True	True
True	False	False
False	True	False
False	False	True

Well Formed Formula

Well Formed Formula (wff) is a predicate holding one of the following −

• All propositional constants and propositional variables are wffs.
• If x is a variable and Y is a wff, ∀xY and ∃xY are also wff.
• Truth value and false values are wffs.
• Each atomic formula is a wff.
• All connectives connecting wffs are wffs.

Quantifiers

The variable of predicates is quantified by quantifiers. There are two types of quantifier in predicate logic −

• Universal Quantifier
• Existential Quantifier

Universal Quantifier

Universal quantifier states that the statements within its scope are true for every value of the specific variable. It is denoted by the symbol ∀.

∀xP(x) is read as for every value of x, P(x) is true.

Example − "Man is mortal" can be transformed into the propositional form ∀xP(x). Here, P(x) is the predicate which denotes that x is mortal and the universe of discourse is all men.

Existential Quantifier

Existential quantifier states that the statements within its scope are true for some values of the specific variable. It is denoted by the symbol ∃.

∃xP(x) for some values of x is read as, P(x) is true.

Example − "Some people are dishonest" can be transformed into the propositional form ∃x P(x) where P(x) is the predicate which denotes x is dishonest and the universe of discourse is some people.

Nested Quantifiers

If we use a quantifier that appears within the scope of another quantifier, it is called a nested quantifier.

Example

• ∀ a∃bP(x,y) where P(a,b) denotes a+b = 0
• ∀ a∀b∀cP(a,b,c) where P(a,b) denotes a+(b+c) = (a+b)+c

Membership Function

 

Membership Function

We already know that fuzzy logic is not logic that is fuzzy but logic that is used to describe fuzziness. This fuzziness is best characterized by its membership function. In other words, we can say that membership function represents the degree of truth in fuzzy logic.

Following are a few important points relating to the membership function −

• Membership functions were first introduced in 1965 by Lofti A. Zadeh in his first research paper “fuzzy sets”.

• Membership functions characterize fuzziness (i.e., all the information in fuzzy set), whether the elements in fuzzy sets are discrete or continuous.

• Membership functions can be defined as a technique to solve practical problems by experience rather than knowledge.

• Membership functions are represented by graphical forms.

• Rules for defining fuzziness are fuzzy too.

Mathematical Notation

We have already studied that a fuzzy set Ã in the universe of information Ucan be defined as a set of ordered pairs and it can be represented mathematically as −

A˜={(y,μA˜(y))|y∈U}A~={(y,μA~(y))|y∈U}

Here μA˜(∙)μA~(∙) = membership function of A˜A~; this assumes values in the range from 0 to 1, i.e., μA˜(∙)∈[0,1]μA~(∙)∈[0,1]. The membership function μA˜(∙)μA~(∙) maps UU to the membership spaceMM.

The dot (∙)(∙) in the membership function described above, represents the element in a fuzzy set; whether it is discrete or continuous.

Features of Membership Functions

We will now discuss the different features of Membership Functions.

Core

For any fuzzy set A˜A~, the core of a membership function is that region of universe that is characterize by full membership in the set. Hence, core consists of all those elements yy of the universe of information such that,

μA˜(y)=1μA~(y)=1

Support

For any fuzzy set A˜A~, the support of a membership function is the region of universe that is characterize by a nonzero membership in the set. Hence core consists of all those elements yy of the universe of information such that,

μA˜(y)>0μA~(y)>0

Boundary

For any fuzzy set A˜A~, the boundary of a membership function is the region of universe that is characterized by a nonzero but incomplete membership in the set. Hence, core consists of all those elements yy of the universe of information such that,

1>μA˜(y)>01>μA~(y)>0

Fuzzification

It may be defined as the process of transforming a crisp set to a fuzzy set or a fuzzy set to fuzzier set. Basically, this operation translates accurate crisp input values into linguistic variables.

Following are the two important methods of fuzzification −

Support Fuzzification(s-fuzzification) Method

In this method, the fuzzified set can be expressed with the help of the following relation −

A˜=μ1Q(x1)+μ2Q(x2)+...+μnQ(xn)A~=μ1Q(x1)+μ2Q(x2)+...+μnQ(xn)

Here the fuzzy set Q(xi)Q(xi) is called as kernel of fuzzification. This method is implemented by keeping μiμi constant and xixi being transformed to a fuzzy set Q(xi)Q(xi).

Grade Fuzzification (g-fuzzification) Method

It is quite similar to the above method but the main difference is that it kept xixiconstant and μiμi is expressed as a fuzzy set.

Defuzzification

It may be defined as the process of reducing a fuzzy set into a crisp set or to convert a fuzzy member into a crisp member.

We have already studied that the fuzzification process involves conversion from crisp quantities to fuzzy quantities. In a number of engineering applications, it is necessary to defuzzify the result or rather “fuzzy result” so that it must be converted to crisp result. Mathematically, the process of Defuzzification is also called “rounding it off”.

The different methods of Defuzzification are described below −

Max-Membership Method

This method is limited to peak output functions and also known as height method. Mathematically it can be represented as follows −

μA˜(x∗)>μA˜(x)forallx∈XμA~(x∗)>μA~(x)forallx∈X

Here, x∗x∗ is the defuzzified output.

Centroid Method

This method is also known as the center of area or the center of gravity method. Mathematically, the defuzzified output x∗x∗ will be represented as −

x∗=∫μA˜(x).xdx∫μA˜(x).dxx∗=∫μA~(x).xdx∫μA~(x).dx

Weighted Average Method

In this method, each membership function is weighted by its maximum membership value. Mathematically, the defuzzified output x∗x∗ will be represented as −

x∗=∑μA˜(xi¯¯¯¯¯).xi¯¯¯¯¯∑μA˜(xi¯¯¯¯¯)x∗=∑μA~(xi¯).xi¯∑μA~(xi¯)

Mean-Max Membership

This method is also known as the middle of the maxima. Mathematically, the defuzzified output x∗x∗ will be represented as −

x∗=∑i=1nxi¯¯

By pmH bn C

Set Theory

 

Set Theory

Fuzzy sets can be considered as an extension and gross oversimplification of classical sets. It can be best understood in the context of set membership. Basically it allows partial membership which means that it contain elements that have varying degrees of membership in the set. From this, we can understand the difference between classical set and fuzzy set. Classical set contains elements that satisfy precise properties of membership while fuzzy set contains elements that satisfy imprecise properties of membership.

Mathematical Concept

A fuzzy set A˜A~ in the universe of information UU can be defined as a set of ordered pairs and it can be represented mathematically as −

A˜={(y,μA˜(y))|y∈U}A~={(y,μA~(y))|y∈U}

Here μA˜(y)μA~(y) = degree of membership of yy in \widetilde{A}, assumes values in the range from 0 to 1, i.e., μA˜(y)∈[0,1]μA~(y)∈[0,1].

Representation of fuzzy set

Let us now consider two cases of universe of information and understand how a fuzzy set can be represented.

Case 1

When universe of information UU is discrete and finite −

A˜={μA˜(y1)y1+μA˜(y2)y2+μA˜(y3)y3+...}A~={μA~(y1)y1+μA~(y2)y2+μA~(y3)y3+...}

={∑ni=1μA˜(yi)yi}={∑i=1nμA~(yi)yi}

Case 2

When universe of information UU is continuous and infinite −

A˜={∫μA˜(y)y}A~={∫μA~(y)y}

In the above representation, the summation symbol represents the collection of each element.

Operations on Fuzzy Sets

Having two fuzzy sets A˜A~ and B˜B~, the universe of information UU and an element 𝑦 of the universe, the following relations express the union, intersection and complement operation on fuzzy sets.

Union/Fuzzy ‘OR’

Let us consider the following representation to understand how the Union/Fuzzy ‘OR’ relation works −

μA˜∪B˜(y)=μA˜∨μB˜∀y∈UμA~∪B~(y)=μA~∨μB~∀y∈U

Here ∨ represents the ‘max’ operation.

Intersection/Fuzzy ‘AND’

Let us consider the following representation to understand how the Intersection/Fuzzy ‘AND’ relation works −

μA˜∩B˜(y)=μA˜∧μB˜∀y∈UμA~∩B~(y)=μA~∧μB~∀y∈U

Here ∧ represents the ‘min’ operation.

Complement/Fuzzy ‘NOT’

Let us consider the following representation to understand how the Complement/Fuzzy ‘NOT’ relation works −

μA˜=1−μA˜(y)y∈UμA~=1−μA~(y)y∈U

Properties of Fuzzy Sets

Let us discuss the different properties of fuzzy sets.

Commutative Property

Having two fuzzy sets A˜A~ and B˜B~, this property states −

A˜∪B˜=B˜∪A˜A~∪B~=B~∪A~

A˜∩B˜=B˜∩A˜A~∩B~=B~∩A~

Associative Property

Having three fuzzy sets A˜A~, B˜B~ and C˜C~, this property states −

A˜∪(B˜∪C˜)=(A˜∪B˜)∪C˜A~∪(B~∪C~)=(A~∪B~)∪C~

A˜∩(B˜∩C˜)=(A˜∪B˜)∪C˜A~∩(B~∩C~)=(A~∪B~)∪C~

Distributive Property

Having three fuzzy sets A˜A~, B˜B~ and C˜C~, this property states −

A˜∪(B˜∩C˜)=(A˜∪B˜)∩(A˜∪C˜)A~∪(B~∩C~)=(A~∪B~)∩(A~∪C~)

A˜∩(B˜∪C˜)=(A˜∩B˜)∪(A˜∩C˜)A~∩(B~∪C~)=(A~∩B~)∪(A~∩C~)

Idempotency Property

For any fuzzy set A˜A~, this property states −

A˜∪A˜=A˜A~∪A~=A~

A˜∩A˜=A˜A~∩A~=A~

Identity Property

For fuzzy set A˜A~ and universal set UU, this property states −

A˜∪φ=A˜A~∪φ=A~

A˜∩U=A˜A~∩U=A~

A˜∩φ=φA~∩φ=φ

A˜∪U=UA~∪U=U

Transitive Property

Having three fuzzy sets A˜A~, B˜B~ and C˜C~, this property states −

IfA˜⊆B˜⊆C˜,thenA˜⊆C˜IfA~⊆B~⊆C~,thenA~⊆C~

Involution Property

For any fuzzy set A˜A~, this property states −

A˜¯¯¯¯¯¯¯¯=A˜A~¯¯=A~

De Morgan’s Law

This law plays a crucial role in proving tautologies and contradiction. This law states −

A˜∩B˜¯¯¯¯¯¯¯¯¯¯¯¯¯=A˜¯¯¯¯∪B˜¯¯¯¯A~∩B~¯=A~¯∪B~¯

By Prof  Sanjeev kumar

     Er Ashish AC Nishad

Classical Set Theory

 

Classical Set Theory

A set is an unordered collection of different elements. It can be written explicitly by listing its elements using the set bracket. If the order of the elements is changed or any element of a set is repeated, it does not make any changes in the set.

Example

• A set of all positive integers.
• A set of all the planets in the solar system.
• A set of all the states in India.
• A set of all the lowercase letters of the alphabet.

Mathematical Representation of a Set

Sets can be represented in two ways −

Roster or Tabular Form

In this form, a set is represented by listing all the elements comprising it. The elements are enclosed within braces and separated by commas.

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

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

Set Builder Notation

In this form, the set is defined by specifying a property that elements of the set have in common. The set is described as A = {x:p(x)}

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

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

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

B = {x:1 ≤ x < 10 and (x%2) ≠ 0}

If an element x is a member of any set S, it is denoted by x∈S and if an element y is not a member of set S, it is denoted by y∉S.

Example − If S = {1,1.2,1.7,2},1 ∈ S but 1.5 ∉ S

Cardinality of a Set

Cardinality of a set S, denoted by |S||S|, is the number of elements of the set. The number is also referred as the cardinal number. If a set has an infinite number of elements, its cardinality is ∞∞.

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

If there are two sets X and Y, |X| = |Y| denotes two sets X and Y having same cardinality. It occurs when the number of elements in X is exactly equal to the number of elements in Y. In this case, there exists a bijective function ‘f’ from X to Y.

|X| ≤ |Y| denotes that set X’s cardinality is less than or equal to set Y’s cardinality. It occurs when the number of elements in X is less than or equal to that of Y. Here, there exists an injective function ‘f’ from X to Y.

|X| < |Y| denotes that set X’s cardinality is less than set Y’s cardinality. It occurs when the number of elements in X is less than that of Y. Here, the function ‘f’ from X to Y is injective function but not bijective.

If |X| ≤ |Y| and |X| ≤ |Y| then |X| = |Y|. The sets X and Y are commonly referred as equivalent sets.

Types of Sets

Sets can be classified into many types; some of which are finite, infinite, subset, universal, proper, singleton set, etc.

Finite Set

A set which contains a definite number of elements is called a finite set.

Example − S = {x|x ∈ N and 70 > x > 50}

Infinite Set

A set which contains infinite number of elements is called an infinite set.

Example − S = {x|x ∈ N and x > 10}

Subset

A set X is a subset of set Y (Written as X ⊆ Y) if every element of X is an element of set Y.

Example 1 − Let, X = {1,2,3,4,5,6} and Y = {1,2}. Here set Y is a subset of set X as all the elements of set Y is in set X. Hence, we can write Y⊆X.

Example 2 − Let, X = {1,2,3} and Y = {1,2,3}. Here set Y is a subset (not a proper subset) of set X as all the elements of set Y is in set X. Hence, we can write Y⊆X.

Proper Subset

The term “proper subset” can be defined as “subset of but not equal to”. A Set X is a proper subset of set Y (Written as X ⊂ Y) if every element of X is an element of set Y and |X| < |Y|.

Example − Let, X = {1,2,3,4,5,6} and Y = {1,2}. Here set Y ⊂ X, since all elements in Y are contained in X too and X has at least one element which is more than set Y.

Universal Set

It is a collection of all elements in a particular context or application. All the sets in that context or application are essentially subsets of this universal set. Universal sets are represented as U.

Example − We may define U as the set of all animals on earth. In 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 insects is a subset of U, and so on.

Empty Set or Null Set

An empty set contains no elements. It is denoted by Φ. As the number of elements in an empty set is finite, empty set is a finite set. The cardinality of empty set or null set is zero.

Example – S = {x|x ∈ N and 7 < x < 8} = Φ

Singleton Set or Unit Set

A Singleton set or Unit set contains only one element. A singleton set is denoted by {s}.

Example − S = {x|x ∈ N, 7 < x < 9} = {8}

Equal Set

If two sets contain the same elements, they are said to be equal.

Example − If A = {1,2,6} and B = {6,1,2}, they are equal as every element of set A is an element of set B and every element of set B is an element of set A.

Equivalent Set

If the cardinalities of two sets are same, they are called equivalent sets.

Example − If A = {1,2,6} and B = {16,17,22}, they are equivalent as cardinality of A is equal to the cardinality of B. i.e. |A| = |B| = 3

Overlapping Set

Two sets that have at least one common element are called overlapping sets. In case of overlapping sets −

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

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

n(A)=n(A−B)+n(A∩B)n(A)=n(A−B)+n(A∩B)

n(B)=n(B−A)+n(A∩B)n(B)=n(B−A)+n(A∩B)

Example − Let, A = {1,2,6} and B = {6,12,42}. There is a common element ‘6’, hence these sets are overlapping sets.

Disjoint Set

Two sets A and B are called disjoint sets if they do not have even one element in common. Therefore, disjoint sets have the following properties −

n(A∩B)=ϕn(A∩B)=ϕ

n(A∪B)=n(A)+n(B)n(A∪B)=n(A)+n(B)

Example − Let, A = {1,2,6} and B = {7,9,14}, there is not a single common element, hence these sets are overlapping sets.

Operations on Classical Sets

Set Operations include Set Union, Set Intersection, Set Difference, Complement of Set, and Cartesian Product.

Union

The union of sets A and B (denoted by A ∪ BA ∪ B) is the set of elements which are in A, in B, or in both A and B. Hence, A ∪ B = {x|x ∈ A OR x ∈ B}.

Example − If A = {10,11,12,13} and B = {13,14,15}, then A ∪ B = {10,11,12,13,14,15} – The common element occurs only once.

Intersection

The intersection of sets A and B (denoted by A ∩ B) is the set of elements which are in both A and B. Hence, A ∩ B = {x|x ∈ A AND x ∈ B}.

Difference/ Relative Complement

The set difference of sets A and B (denoted by A–B) is the set of elements which are only in A but not in B. Hence, A − B = {x|x ∈ A AND x ∉ B}.

Example − If A = {10,11,12,13} and B = {13,14,15}, then (A − B) = {10,11,12} and (B − A) = {14,15}. Here, we can see (A − B) ≠ (B − A)

Complement of a Set

The complement of a set A (denoted by A′) is the set of elements which are not in set A. Hence, A′ = {x|x ∉ A}.

More specifically, A′ = (U−A) where U is a universal set which contains all objects.

Example − If A = {x|x belongs to set of add integers} then A′ = {y|y does not belong to set of odd integers}

Cartesian Product / Cross Product

The Cartesian product of n number of sets A1,A2,…An denoted as A1 × A2...× An can be defined as all possible ordered pairs (x1,x2,…xn) where x1 ∈ A1,x2 ∈ A2,…xn ∈ An

Example − If we take two sets 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 Sets

Properties on sets play an important role for obtaining the solution. Following are the different properties of classical sets −

Commutative Property

Having two sets A and B, this property states −

A∪B=B∪AA∪B=B∪A

A∩B=B∩AA∩B=B∩A

Associative Property

Having three sets A, B and C, this property states −

A∪(B∪C)=(A∪B)∪CA∪(B∪C)=(A∪B)∪C

A∩(B∩C)=(A∩B)∩CA∩(B∩C)=(A∩B)∩C

Distributive Property

Having three sets A, B and C, this property states −

A∪(B∩C)=(A∪B)∩(A∪C)A∪(B∩C)=(A∪B)∩(A∪C)

A∩(B∪C)=(A∩B)∪(A∩C)A∩(B∪C)=(A∩B)∪(A∩C)

Idempotency Property

For any set A, this property states −

A∪A=AA∪A=A

A∩A=AA∩A=A

Identity Property

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

A∪φ=AA∪φ=A

A∩X=AA∩X=A

A∩φ=φA∩φ=φ

A∪X=XA∪X=X

Transitive Property

Having three sets A, B and C, the property states −

If A⊆B⊆CA⊆B⊆C, then A⊆CA⊆C

Involution Property

For any set A, this property states −

A¯¯¯¯¯¯¯¯=AA¯¯=A

De Morgan’s Law

It is a very important law and supports in proving tautologies and contradiction. This law states −

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

A∪B¯¯¯¯¯¯¯¯¯¯¯¯¯=A¯¯¯¯∩B¯¯¯¯