Maths and Logic

Question 1

what is a field?

Question 2

what are the laws?

Answer 2

https://qph.cf2.quoracdn.net/main-qimg-acaef512c516464f577170e2c55f02e2-lq

Question 3

what is a commutative semiring and a ring?

Question 4

what is an axiom schemata?

Question 5

what are the steps of deriving logical expressions?

Question 6

what does ambiguous mean?

Answer 6

order of operations not specified such as a -> b -> c
could mean (a -> b) -> c OR a -> (b -> c)

Question 7

what is the difference between classical and constructive deduction?

Question 8

what is the double negation law?

Answer 8

¬¬A ⊢ A

Question 9

what is the law of excluded middle

Answer 9

⊢ A ∨ ¬A

Question 10

do flashcard on how vertical proof thing works

Question 11

what is the symbol for ‘is a member of’ and ‘is a subset of’?

Answer 11

∈ and ⊆

Question 12

how do you write the items in A that aren’t in B?

Answer 12

A\B

Question 13

what does it mean to prove something constructively?

Answer 13

no classical reasoning is used (DNE and LEM)

Question 14

What is an endorelation and an endofunction?

Answer 14

A relation/ function relating items in a set to other items in the same set

Question 15

What is an equivalence relation?

Answer 15

A relation that is reflexive, symmetric and transitive

Question 16

What does reflexive, symmetric and transitive mean?

Answer 16

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

Question 17

What is an equivalence class?

Answer 17

All elements grouped together that relate to the same element

Question 18

How do you write the equivalence class of X and the set of all equivalence classes?

Answer 18

answer in week 7 materials

Question 19

What is a partition of a set?

Answer 19

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

Question 20

How does an equivalence relation correspond to a partition?

Answer 20

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

Question 21

What is a kernel?

Answer 21

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

Question 22

What does irreflexive and antisymmetric mean?

Answer 22

Irreflexive: no elements relate to themselves
Antisymmetric: the only way two elements can be mutually related is if they are equal

Question 23

What is an order?