CST 329 - Week 7

This was our seventh week in CST 329 - Reasoning with Logic.

Chapter 15: Relations, Functions, Identity, Multiple Quantifiers

Relation: A predicate with an arity of greater than one. Such predicates relate pairs of things in the domain of discourse. They can also be thought of as sets of ordered things from the domain of discourse where said predicate is true. 

Reflexive Relations: For any relation Φ(xy), the relation is reflexive if and only if ∀xΦ(xx).
 
Symmetric Relations: For any relation Φ, the relation is symmetric if and only if ∀x∀y(Φ(x y) → Φ(y x)).
 
Transitive Relations: Finally, call a relation “transitive” if and only if ∀x∀y∀z((Φ(x y)^Φ(y z)) → Φ(x z))
 
Function: A predicate relates a specific number of things to one thing. Addition is a function, it takes two numbers and relates them to one other number. A function is represented by f, g, h... and a number of symbolic terms equal to its arity. So fx is a function.
 
Translation key with function fx
 
Identity: A special type of predicate that states that one thing is equal to another thing. Can be written as x=y or a two arity predicate such as Ixy.
 

Indiscernibility of Identicals

or Substitution of Identicals

 
Multiple Quantifiers: In statements with multiple quantifiers, we read the quantifiers from left to right.
  • ∀x∃y x≥y: Pick any number in our domain of discourse, then there will be at least one number in our domain of discourse that is less than or equal to that first number that you already picked.
  • ∃y∀x x≥y: There is at least one number in our domain of discourse such that, if you pick that number, then any number in our domain of discourse is greater than or equal to it.
 
Sentences translated to FOL sentences with multiple quantifiers

Chapter 16: Summary of FOL

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