answers.org

category theory for the sciences

https://mitpress.mit.edu/books/category-theory-sciences

exercises

2.1.2.3

a) PR → RG b) no , i think that in the brain one cell could connect to multiple cells in the codomain

2.1.2.4

a) 2 ↦ 4 b) 0 ↦ 0 c) -2 ↦ 4 d) 5 ↦ 25 e)

• → denotes an arrow that describes a function from the domain set to the codomain set,
• ↦ denotes an mapping from one value directly to another

2.1.2.6

im(f) = {y₁,y₂,y₄}

2.1.2.8

what is the image set of f(A)? well A is given as A := {-1, 0, 1, 2, 3} which can be seen as fₐ := x ↦ x = id, so we can compose it in our comprehension got the answer wrong initially , this states the set of things such that there is a function now as i ⊆ X can be seen as a function i → x we can treat the subset as a function and state

i(a) composed with f that equals x, where x is from ℤ f(A) = { x ∈ ℤ | ∃a ∈ A such that f · i(a) = x}

2.1.2.10

y : {:)} → Y

2.1.2.12

a) 2⁵ = 32 b) 5² = 25

2.1.2.13

a) ∅ b) {1}

should be

a) {x ∈ any Set} b) ∅

to have homSet(X,A) == to 1 for all A then there only needs to be the function that maps everything to one value it must be a function that maps everthing to one member of a set, therefore A; the codomain must be aset of one element

to have homSet(B,X) == 1 for all B , B must be the empty set since there can only be one value for the empty set

2.1.2.17

a) n! b) yes

2.1.2.19

a) no

2.1.2.20

a) because the function id can be seen as x ⊆ X then , we can say the single element set , is the same as asserting membership of X , because is function in i → i points to one element, then it is an isomorphism

2.1.2.22

a) c b) i:= {1,4,9,16,25,36,49}

2.1.2.24

a) 3 b) not a set c) infinite d) 6