diff --git a/_chapters/01_set.md b/_chapters/01_set.md index 4ebffdf..68c6218 100644 --- a/_chapters/01_set.md +++ b/_chapters/01_set.md @@ -68,7 +68,7 @@ Of course if one is a valid answer, so can be zero. If we want a set of all *bla Note that a set is defined only by the items it contains, which means that there is no difference between the set that contains zero *balls* and the set that contains zero *numbers*, for instance. In other words, the empty set is *unique* set, which makes it a very special one. Formally, the empty set is marked with the symbol **∅** (so **B = W = ∅**). -The empty set is a special one, for example, it is a subset of every other set (mathematically speaking, **∀ A \| A ⊆ ∅**) +The empty set is a special one, for example, it is a subset of every other set (mathematically speaking, **∀ A \| ∅ ⊆ A**) We will encounter the empty set again. @@ -286,7 +286,7 @@ The Power of Composition To understand how powerful composition is, consider the following: one set being connected to another means that each function from the second set can be transferred to a corresponding function from the first one. -If we have a function **g: P → Y ** from set **P** to set **Y**, then for every function **f** from the set **Y** to any other set, there is a corresponding function **f ∘ g** from the set **P** to the same set. In other words, every time you define a new function from **Y** to some other set, you gain one function from **P** to that same set for free. +If we have a function **g: P → Y** from set **P** to set **Y**, then for every function **f** from the set **Y** to any other set, there is a corresponding function **f ∘ g** from the set **P** to the same set. In other words, every time you define a new function from **Y** to some other set, you gain one function from **P** to that same set for free. ![Functional composition connect](morphism_general.svg) diff --git a/_chapters/02_category/coproduct_product_duality.svg b/_chapters/02_category/coproduct_product_duality.svg index 33606c0..b165bbd 100644 --- a/_chapters/02_category/coproduct_product_duality.svg +++ b/_chapters/02_category/coproduct_product_duality.svg @@ -1 +1,252 @@ -ANDOR \ No newline at end of file + + + + + + image/svg+xml + + + + + + + + + + + + + + + + + + + + + + + + + + AND + OR + diff --git a/_chapters/04_order.md b/_chapters/04_order.md index 39d3ae4..4f7f7e8 100644 --- a/_chapters/04_order.md +++ b/_chapters/04_order.md @@ -60,7 +60,7 @@ This is the law that to a large extend defines what an order is: if I am better Antisymmetry --- -The third law is called antisymmetry and it states that the function that defines the order should not give contradictory results (**a ≤ b ⟺ b ≰ a**). +The third law is called antisymmetry and it states that the function that defines the order should not give contradictory results (or in other words you have **x ≤ y** and **y ≤ x** only if **x = y**). ![antisymmetry](antisymmetry.svg) @@ -71,7 +71,7 @@ Totality The last law is called *totality* (or *connexity*) and it mandates that all elements that belong to the order should be comparable - **a ≤ b or b ≤ a**. That is, for any two elements, one would always be "bigger" than the other. -By the way, this law makes the reflexivity law redundant, as it is just a special case of reflexivity when **a** and **b** are one and the same object, but I still want to present it for reasons that will become apparent soon. +By the way, this law makes the reflexivity law redundant, as reflexivity is just a special case of totality when **a** and **b** are one and the same object, but I still want to present it for reasons that will become apparent soon. ![connexity](connexity.svg) @@ -171,7 +171,7 @@ Like with the maximum element, if two elements have several upper bounds that ar ![A non-join diagram](non_join.svg) -If, however, one of those elements is established as bigger than another, it immediately qualifies. +If, however, one of those elements is established as smaller than the rest of them, it immediately qualifies. ![A join diagram](non_join_fix.svg) diff --git a/dictionary.txt b/dictionary.txt new file mode 100644 index 0000000..26cd77d --- /dev/null +++ b/dictionary.txt @@ -0,0 +1,49 @@ +coproduct +morphism +coproducts +morphisms +preorders +Preorders +antisymmetry +antisymmetric +preorder +Preorder +semilattice +Semilattice +semilattices +Semilattices +meet-semilattice +meet-semilattices +poset +posets +Posets +Birkhoff's +superset +monoid +monoid-like +monoids +monoidal +monoid's +Monoid +Monoids +isomorphism +isomorphisms +Hasse +Antisymmetry +connexity +linearly +Heyting +modus +Modus +ponens +intuitionistic +leit +Z3 +Z2 +Z1 +abelian +Abelian +non-abelian +Dih3 +composable +forall