Easy Math Learning: Set Theory

Introduction to Logic and Set Theory Learning:

Logic theory is a set of sentences in a formal language. The individual sentences of a theory are called as the theorems. A first-order theory is a set of first-order sentences. Many authors require that the theory be closed under logical consequence; a theory with this property can be called a deductive theory. Set theory is the studies of sets, which are collections of objects. Although any type of object can be collected into a set.

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Basic operations of logic and set theory learning

Union in Set theory Learning:
If A and B are different sets then union of A and B denoted as A U B.

For example {3, 4, 5} and {6, 7, 8} is the set {3, 4, 5, 6, 7, 8.

Intersection in Set theory Learning:
If A and B are different sets then intersection of A and B denoted as A n B.

For example {3, 4, 5} and {6, 4, 3} is the set {3, 4}.

Complement in Set theory Learning:
If A and B are different sets then A relative to set b, denoted Ac.

For example complement of {3, 4, 5} relative to {6, 4, 3} is {6}.

Symmetric difference in Set theory Learning:
If A and B are different sets then member of exactly one of A and B.

For example complement of {3, 4, 5} relative to {6, 4, 5} is {3, 6}.

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logic and set theory learning

Negation (NOT) ˜ p in logic learning:
If value of proposition is true then transform into false and vice versa.

Disjunction (OR) p v q in logic learning:
If two propositions are false then the result is false otherwise true.

Conjunction (AND) p ^ q in logic learning:
If two propositions are true then result is true otherwise false.

Conditional (IF…THEN…) p?q in logic learning :
Truth of the proposition p is sufficient to truth of proposition q.

Biconditional (IF AND ONLY IF) p?q in logic learning:
p is sufficient condition for q. q is necessary for p. Unless q not p. Not p unless q. Not p without q.

Introduction to descriptive set theory:
A set is a well-defined collection of things

The things belonging to a set are called it's members or elements. These elements may be objects, persons, letters, numbers or any items at all.
Well-defined means that given a member, there must be no doubt in deciding whether it belongs to, or does not belong to the given set. Example:-
A set of apples weighting 200 g or over is a well-defined.
A collection of heavy apples is not well-defined and so it not a set.
All elements are separated by a comma.
The elements should be enclosed in braces (”{ }”).
Sets are usually represented by capital letters.
The order in which the members appear is not important.
If one or more elements are repeated, the set remains the same.
A set may consist of a single member like, B = {National bird of India} = {Peacock}. If 'A' is a set and 'a' is an elements of this set, then “a belongs to A”
This article is about descriptive set theory and the topics in that.

Descriptive Set Theory-representation of a Set (notation):

A set is determined by its members or elements. This determination is brought out in the following three ways.

Roster method :-

The method of listing the elements inside the braces is called Roster method. Example :- A set of alphabets of the English language represented by P is written as :
P = {A , B , C , D , E , F , ….}

Description method :-

The method of listing a set by a well-defined statement or description is called description method.
Example :- {Vowels in English alphabet}

Set builder form :-

The method of listing the elements inside the braces by stating the rule or property or formula is known as Set builder form. Example :- If A is a set of a natural numbers less than 80, then it can be represented as, A = {x : x is a natural number and is less than 80} or as

A = {=x : x E N, x<80 br="br">
Types of Sets in Descriptive Set Theory:

Finite set

The elements are limited. Example :-
Natural number less then 60.
A = {1,2,3,4,5,6,7,.....}
Days of a week
W = {Monday, ….. , Sunday}

In finite set

Unlimited numbers of elements Example :-
Set of odd numbers
N = {1 , 3 , 5 , 7....}
Set of even numbers

E = {2 , 4 , 6 , 8....}

Empty set

No elements is represented
Set of odd numbers between 5 and 6
A = { }
30th day of February

D = { }