Explain what is set theory 2021

What Is set theory

set theory is a basic building block for types of objects in discrete mathematics.

Set operations in computer programming languages: Issues about data

• structures which is used to represent sets and the computational cost of set operations.
• set theory is a foundation of mathematics.
• Several different systems of axioms have been proposed.
• The standard set theory is Zermelo-Fraenkel set theory (ZF).
• Often extended by the axiom of choice to ZFC.
• Here we can not concerned with a simple set of axioms for set theory. Instead of this, we can use naïve set theory.

Sets

• The set is an unordered collection of objects
• example - students in this class; air molecules in this room.
• The objects in the set is known as elements, or members of set.
• A set is said to consist its elements.
• If x is not a member of A, write x ∈/ A.

Describing a Set: Roster Method

• Every distinct object is either a member or not; listing more than
• once doesn't change the set. A = {a, b, c, d, e, f} = {a, b, c, d, e, f, b, c, d}.
• Dots “. . . . ” is used to define a set without listing all of members.
• when a pattern is clear. A = {a, b, c, d, e, f, . . . , z} or
• A = {3, 4, 5, 6, 7, 8, 9, 10, . . . , 20}.
• Don't overuse this.
• Patterns are not always as clear as a writer thinks.

Some Important Sets

• B = Boolean values = {true or false}
• Z = integers = {. . . , −4, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, . . . }
• Z+ = Z≥1 = positive integers = {1, 2, 3, 4, 5, 6. . ... }
• R = It is set of real numbers.
• R+ = 0<R = set of positive real numbers.
• C = set of complex numbers.
• Q = set of rational numbers.

Set Builder Notation

Define the property / properties that all members of set must define.

• A = {x | x is the positive integer less than 200}
• A = {x | x ∈ Z + ∧ x < 100}.
• S = {x ∈ Z + | x < 100}
• A predicate can be used.
• Example : S = {x | P(x)}
• Where P(x) is true if x is prime number.

Positive rational numbers

Q+ = {x ∈ R | ∃p, q ∈ Z+ x = p/q}.

Interval Notation

Used to define subsets of sets upon which an order is described, example - numbers.

Universal Set and Empty Set

The universal set is a set including everything currently under consideration.

Universal set is symbolized by U.

• It Content depends on the context.
• It Sometimes explicitly stated, sometimes implicit.

The empty set is a set with null / no elements.

and it is Symbolized by ∅ or {}.