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escape chars for discrete
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avipars authored Mar 20, 2024
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- Cartesian product:

* A × B = { (a, b) | a ∈ A and b ∈ B }
* `A × B = { (a, b) | a ∈ A and b ∈ B }`

* A × B ≠ B × A (unless A = B)

* A ∩ (A × B) = ∅

* |A × B| = |A| × |B|
* `|A × B| = |A| × |B|`

- Distribution:

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* implies that the relation R contains all possible pairs that can be formed by taking one element from set A and one element from set B. It's a relation where every element of A is related to every element of B.

- Inverse relation is R⁻¹ = { (b, a) | (a, b) ∈ R }
- Inverse relation is R⁻¹ = `{ (b, a) | (a, b) ∈ R }`

* R consists of all pairs in R but with their elements reversed. If (a,b) is in R, then (b,a) is in R⁻¹

- Composition of relations:

* R ∘ S = { (a, c) | ∃ b : (a, b) ∈ R and (b, c) ∈ S }
* R ∘ S = `{ (a, c) | ∃ b : (a, b) ∈ R and (b, c) ∈ S }`

* Set of pairs (a,c) such that exists an element b for which both (a,b) is in R and (b,c) is in S

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