# Fuzzy Logic - Classical Set Theory

A ** set ** is an unordered collection of different elements. It can be written explicitly by listing its elements using the defined parenthesis. If the order of the elements is changed or if an element of a set is repeated, this does not make any change to the set.

### Example

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

## Mathematical representation of a set

Sets can be represented in two ways -

### List or tabular form

In this form, a set is represented by listing all the elements that compose it. The items are surrounded by braces and separated by commas.

Seehere examples of sets in list or in 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 the 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} s 'written

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

** Example 2 ** - The set {1,3,5,7,9} is written

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 the set S, it is noted y∉S.

** Example ** - If S = {1,1.2,1.7,2}, 1 ∈ S but 1,5 ∉ S

### Cardinality of a set

The cardinality of a set S, noted | S || S |, is the number of elements in the set. The number is also called a 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 t wo sets X and Y, * | X | = | Y | * designates two sets X and Y having the same cardinality. This happens when the number of elements in X is exactly equal to the number of elements in Y. In this case, there is a one-to-one function "f" from X to Y.

* | X | ≤ | Y | * indicates that the cardinality of set X is less than or equal to the cardinality of set Y. This happens when the number of elements in X is less than or equal to that of Y. Here , there is an injective function "f" from X to Y.

* | X | <| Y | * indicates that the cardinality of set X is less than the cardinality of set Y. This happens when the number of elements in X is lessto that of Y. Here, the function “f” from X to Y is an injective but not a bijective function.

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

## Types of sets

Sets can be classified into several types; some of which are finite, infinite, subsets, universal, proper, singleton, 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 an 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 each element of X is an element of the set Y.

** Example 1 ** - Let X = {1,2,3,4,5,6} and Y ={1,2}. Here the set Y is a subset of the set X because all the elements of the set Y are in the set X. Therefore, we can write Y⊆X.

** Example 2 ** - Let X = {1,2,3} and Y = {1,2,3}. Here, the set Y is a subset (not a proper subset) of the set X because all the elements of the set Y are in the set X. Therefore, 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 the set Y (written as X ⊂ Y) if each element of X is an element of the set Y and | X | <|Y|.

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

### Universal set

It 'sa collection of all the elements in onecontext or a particular application. All sets in this context or application are essentially subsets of this universal set. Universal sets are represented by U.

** Example ** - We can 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 fish 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 noted Φ. Since the number of elements in an empty set is finite, the empty set is a finite set. The cardinality of an empty set or a null set is zero.

** Example ** - S = {x | x ∈ N and 7

### Singleton set or set of units

A singleton set or set of units contains a single element. Anset of singleton is denoted by {s}.

** Example ** - S = {x | x ∈ N, 7

### 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 because each element of set A is an element of set B and each element of set B is an element of set A.

### Equivalent set

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

** Example ** - If A = {1,2,6} and B = {16,17,22}, they are equivalent because the cardinality of A is equal to the cardinality by B. ie | A | = | B | = 3

### Overlapping set

Two sets which have at least one common element are called overlapping sets. In case of overlapping sets -

$$ n left (A cup B right) = n left (A right) + n left (B right) - n left (A cup B right) $ $

$$n left (A cup B right) = n left (AB right) + n left (BA right) + n left (A cap B right) $$

$$ n left (A right) = n left (AB rig ht) + n left (A cap B right) $$

$$ n left (B right) = n left (BA right) + n left (A cap B right) $$

** Example ** - Let, A = {1,2,6} and B = {6,12, 42}. There is a common element '6 ', so these sets are overlapping sets.

### Disjoint set

Two sets A and B are called disjoint sets if they don't even have an element in common. Therefore, disjoint sets have the following properties -

$$ n left (A cap B right) = phi $$

$$ n left (A cup B right) = n left (Right) + n left (B right) $$

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

## Operations on clas setssics

Set operations include set union, set intersection, set difference, set complement, and Cartesian product .

### Union

The union of sets A and B (denoted by A ∪ BA ∪ B) is the set of elements that are in A, in B, or in both in A and B. Therefore, 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 appears only once.

### Intersection

The intersection of sets A and B (denoted by A ∩ B) is the set of elements that are both in A and B. Therefore, 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 that are only in A but not in B. Therefore, 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 A ′) is the set of elements which are not in the set A. Therefore, A ′ = {x | x ∉ A}.

More precisely, A ′ = (U - A) where U is a universal set that contains all objects.

** Example ** - If A = {x | x belongs to the set of integers add} then A ′ = {y | yn 'does not belong to the set of odd integers}

### Cartesian Product / Cross Product

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

** Example ** - If we let us take two sets A = {a, b} and B = {1,2},

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

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

## Properties of classical sets

The properties of sets play an important role in obtaining the solution. different properties of classical sets -

### Commutative property

Having two sets ** A ** and ** B **, this property declares -

$$ A cup B = B cup A $$

$$ A cap B = B cap A $$

### Associative property

Having three sets ** A **, ** B ** and ** C **, this property indicates -

$$ A left cup (B right C cup) = left (A right B cup) C $$

$$ A cap left (B cap C right) = left (A cap B right) cap C $$

### Distributive property

Having three sets ** A **, ** B ** and ** C **, this property indicates -

$$ A cup left (B cap C right) = left (A cup B right) cap left (A cup C right) $$

$$ A left cup (B cup right) = left (A cup B right) left cup (A cup C right) $$

### Property of idempotence

For any set ** A **, this property indicates there -

$$ A cup A = A $$

$$ A cap A = A $$

### Identity property

For set ** A ** and universal set ** X **, this property indicates -

$$ A cup varphi = A $ $

$$ A cap X = A $$

$$ A cap varphi = varphi $$

$$ A cup X = X $$

### Transitive property

Has three sets ** A **, ** B **, and ** C **, the property says -

If $ A subseteq B subseteq C $, then $ A subseteq C $

### Involution property

For any set ** A **, this property indicates -

$$ overline {{overline {A}}} = A $$

### De Morgan's law

This is a very important and supports to prove tautologies and contradictions. This law states -

$$ overline {A cap B} = overline {A} cup overline {B} $$

$$ overline {A cup B} = overline {A } cap overline {B} $$