Maths and Logic Flashcards
what is a field?
what are the laws?
https://qph.cf2.quoracdn.net/main-qimg-acaef512c516464f577170e2c55f02e2-lq
what is a commutative semiring and a ring?
what is an axiom schemata?
what are the steps of deriving logical expressions?
what does ambiguous mean?
order of operations not specified such as a -> b -> c
could mean (a -> b) -> c OR a -> (b -> c)
what is the difference between classical and constructive deduction?
what is the double negation law?
¬¬A ⊢ A
what is the law of excluded middle
⊢ A ∨ ¬A
do flashcard on how vertical proof thing works
what is the symbol for ‘is a member of’ and ‘is a subset of’?
∈ and ⊆
how do you write the items in A that aren’t in B?
A\B
what does it mean to prove something constructively?
no classical reasoning is used (DNE and LEM)
What is an endorelation and an endofunction?
A relation/ function relating items in a set to other items in the same set
What is an equivalence relation?
A relation that is reflexive, symmetric and transitive
What does reflexive, symmetric and transitive mean?
Reflexive: every element relates to itself
Symmetric: if X relates to y, y relates to X
Transitive: if X relates to y and y relates to z, X relates to z
What is an equivalence class?
All elements grouped together that relate to the same element
How do you write the equivalence class of X and the set of all equivalence classes?
answer in week 7 materials
What is a partition of a set?
When a set A is split into a set B of subsets of A such that:
-Any two members of B are disjoint
-Any element of A belongs to a member of B
How does an equivalence relation correspond to a partition?
Sets with equivalence relations can always make a partition with the equivalence classes and a partitioned set can always make a set with an equivalence relation
What is a kernel?
Every FUNCTION has a kernel which relates all elements that under the function, have the same image. It is an equivalence relation and the set of all kernels corresponds to the range of the function
What does irreflexive and antisymmetric mean?
Irreflexive: no elements relate to themselves
Antisymmetric: the only way two elements can be mutually related is if they are equal
What is an order?
