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@ -68,7 +68,7 @@ Of course if one is a valid answer, so can be zero. If we want a set of all *bla
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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 = ∅**).
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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 = ∅**).
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The empty set is a special one, for example, it is a subset of every other set (mathematically speaking, **∀ A \| A ⊆ ∅**)
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The empty set is a special one, for example, it is a subset of every other set (mathematically speaking, **∀ A \| ∅ ⊆ A**)
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We will encounter the empty set again.
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We will encounter the empty set again.
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@ -286,7 +286,7 @@ The Power of Composition
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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.
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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.
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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.
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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.
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![Functional composition connect](morphism_general.svg)
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![Functional composition connect](morphism_general.svg)
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@ -60,7 +60,7 @@ This is the law that to a large extend defines what an order is: if I am better
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Antisymmetry
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Antisymmetry
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---
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---
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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**).
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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**).
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![antisymmetry](antisymmetry.svg)
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![antisymmetry](antisymmetry.svg)
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@ -71,7 +71,7 @@ Totality
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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.
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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.
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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.
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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.
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![connexity](connexity.svg)
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![connexity](connexity.svg)
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@ -171,7 +171,7 @@ Like with the maximum element, if two elements have several upper bounds that ar
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![A non-join diagram](non_join.svg)
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![A non-join diagram](non_join.svg)
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If, however, one of those elements is established as bigger than another, it immediately qualifies.
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If, however, one of those elements is established as smaller than the rest of them, it immediately qualifies.
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![A join diagram](non_join_fix.svg)
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![A join diagram](non_join_fix.svg)
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49
dictionary.txt
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49
dictionary.txt
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@ -0,0 +1,49 @@
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coproduct
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morphism
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coproducts
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morphisms
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preorders
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Preorders
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antisymmetry
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antisymmetric
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preorder
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Preorder
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semilattice
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Semilattice
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semilattices
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Semilattices
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meet-semilattice
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meet-semilattices
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poset
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posets
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Posets
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Birkhoff's
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superset
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monoid
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monoid-like
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monoids
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monoidal
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monoid's
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Monoid
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Monoids
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isomorphism
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isomorphisms
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Hasse
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Antisymmetry
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connexity
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linearly
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Heyting
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modus
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Modus
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ponens
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intuitionistic
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leit
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Z3
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Z2
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Z1
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abelian
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Abelian
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non-abelian
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Dih3
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composable
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forall
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