Question

Let A be the set of all functions with domain R and codomain [-1,1].

1. Give 2 functions that are elements of A.
2. Let R = {(f,g) l f(0) = g(0)}. Determine if the given reflection is reflexive, symmetric, and/or transitive. Give brief explanations to go with your answers.
3. Describe in words the set given by [cosx]. (We did a problem before that defined the set [x] for a given relation R by the rule [x] = {y l (x,y) ∈ R}.)

Answer 1

since and cosine come to mind right away

Answer 2

they leave so very quickly...

Answer 3

Thank you for your help; I really appreciate it. I realized I switched the domain and codomain for part a when I initially tried this.

Answer 4

for b, does that mean f and g are each a function where their y intercepts are equal?

Answer 5

Hmm, ya it seems that way :o

Answer 6

then it would definitely be reflexive since the same function would have the same points

Answer 7

Hmm, I'm not sure if I'm missing something...
because showing that it's an equivalence relation almost seems trivial.

Transitive: If fRg then we need gRf.
So if f(0)=g(0),
then g(0)=f(0) is clearly true for any functions in A.

Answer 8

Ya reflexive seems to work out clearly as well.

Answer 9

Thanks :)

Answer 10

For part c, I assumed the set [cosx] just meant the range [-1,1] of the function.. Is that the way to describe that in words?

Answer 11

Hmm I dunno :d the notation is confusing lol.\[\large\rm [\color{orangered}{x}]=\{y~|~(\color{orangered}{x},y)\in R\}\]Based on the previous exercise they referred to,
I guess we have something like this,\[\large\rm [\color{orangered}{\cos x}]=\{y~|~(\color{orangered}{\cos x},y)\in R\}\]So we need ... words?
Like human words for this? Hmmm..

Answer 12

This is the
`set of y values such that cos x is in relation to y`.

That's how I would read it.
So ya, it's uhhh.. the entire range right? as you said? :o

Answer 13

there's probably some pretty way to word that >.<

Answer 14

[cos(x)] is the set of functions f with domain all real numbers and codomain [-1,1] and cos(0)=f(0)

Answer 15

it has been a while since I looked at words codomain and range
I think the range of the function can actually be a subset of the codomain
and they used the word codomain earlier not range

Answer 16

so I think one element of [cos(x)] would be f(x)=1 since cos(0)=f(0)=1
that is just one example what one of the elements I think we would have

Answer 17

also I'm under the assumption we are using the way R was defined in 2...

Answer 18

also circles and ellipses