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# Aggregation: Sets, Relations, Functions

Sets

* A : set, ∀a. a ∈ A, |A| = n, A = {a,b,c,d}
* B : set, ∀b. b ∈ B, |B| = m, B = {1,2,3,4}
* n = m = 4
* cardinality is a unary op, `|A|`
* powerset 𝒫 is a unary op, `𝒫(A)`
* dot product, `⨯`, is a binary op, A⨯B, A⨯A = A²
* number of relations
* number of functions
* number of bijections (1⨯id, inverses) =?= card of A⨯B
* number of injections
* number of surjections
* number of complete collapsors, n → 1

|A| = 4 |B| = 4 |A²| = 4² = 16 |𝒫(A)| = 2⁴ = 16 |𝒫(A²)| = 2¹⁶

Powerset

* 2ⁿ

Cartesian product:

* n⨯m or n²
* card of dot product of 4⁴ = 256
* card of the powerset of dot product = (2ⁿ)²
  * (2⁴)² = 16² = 256 = 2⁸ = 4⁴
* A⨯B = { (a,b) | ∀a∀b. a ∈ A, b ∈ B }
* |A⨯B| = n\*m
* A⨯B ≠ B⨯A
* |A₁⨯B₁| = n₁ ⨯ m₁ = 16
* a set of 16 elements, each one a pair (a₁,b₁)

Relations:

* on finite vs infinite sets
* heterogenous vs homogenous
* have a bunch of properties
* a function is relation with special properties: left-unique, right-serial
* any relation, R, between A and B is a subset of A⨯B
* R = { (a,b) | a ∈ A, b ∈ B }
* 4 → 4 ... there is ? relations

Functions

* 4 → 4 there are 4⁴ = 256 functions
* 4 → 4 there are 16 are bijections (same as card of dot product)

Let two finite sets `A` and `B`, with `|A| = n` and `|B| = m`, also `∀a. a ∈ A` and `∀b. b ∈ B`, then the Carthesian product |A⨯B| = c

A relation, R, on two finite sets, A and B, such that a ∈ A and b ∈ B

The number of distinct relations between A and B: Rᵢ = \[0..k] where k = |A⨯B| R₀ is the empty (null) relation Rₖ is the full (complete) relation, equal to the Cartesian product itself