# Introduction to Set Theory

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

### Set

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 medium.com on September 14, 2017.

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