Introduction to Set Theory

Here’s a summary of things I learned about the Set Theory recently.


A set is a collection into a whole of definite, distinct objects of our intuition or of our thought. The objects are called the elements of the set.

Formal Notation

If a is an element of the set A, we would write it as:a ∈ A

  • 42 ∈ ℕ
  • apple ∈ fruit
  • chicken ∉ mammals

Principle of Extensionality — Set Equivalency

Two sets are equivalent if and only if they contain same objects. Order of elements and the repetition does not affect equivalency.

  • If A = {a} and B = {a} then A = B
  • If A = {a, a} and B = {a} then A = B
  • If A = {a, b, b} and B = {a, b} then A = B
  • If A = {b, a} and B = {a, b} then A = B
  • If A = {a, b} and B = {a} then A ≠ B

Formal Notation

A = B ⇔ ∀x (x ∈ A ⇔ x ∈ B)

Set A is equivalent to set B, if and only if all x where x is an element of set A is also an element of set B

Set Abstraction

Simple sets can be specified fully as shown in examples below.

  • students = {ann, john, james}
  • sports = {footy, tennis, rugby}

While this is sufficient for small sets, a need arises to specify large (or even infinite) sets. Set abstraction allows us to do this using a predicate.

  • odd natural numbers = {n ∈ ℕ | odd(n)}
  • tropical fruits = {fruit ∈ all_fruits| tropical(f)}

Formal Notation

{x | P(x)}

This is a set of all elements that satisfies(or has) the property P.

Originally published at on September 14, 2017.

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