- #1

sa1988

- 222

- 23

## Homework Statement

## Homework Equations

## The Attempt at a Solution

Q.1

I'm a little confused with how subsets and elements are defined in the case of the given set "A" as it seems to be a set of sets, so I'm going to throw my answers out and would appreciate any guidance on where I'm wrong (if anywhere).

[itex]A = \Big\{\emptyset, \big\{\emptyset, \{\emptyset\}\big\}\Big\}[/itex]

"A" has two elements

a) [itex]\emptyset\subset A[/itex] - True

b) [itex]\emptyset\in A [/itex] - True

c) [itex]\{\emptyset\}\subset A [/itex] - Not true

d) [itex]\{\emptyset\}\in A[/itex] - Not true

e) [itex]\{\emptyset, \{\emptyset\}\}\subset A[/itex] - True

f) [itex]\{\emptyset, \{\emptyset\}\}\in A[/itex] - True

Q.2

Not sure what's going on here either. I think the issue is in my own flawed understanding of the notation used in sets generally.

[itex] f : R \rightarrow R[/itex] such that [itex]f(x) = x^{2}[/itex]

My understanding thus far is that the cartesian product of two sets X and Y is:

[itex]X \times Y = \{(x,y) : x\in X, y\in Y\}[/itex]

So in the case of [itex]f(x) = x^2[/itex], we have:

a) [itex]f((-1,2)) = (-1,2) \times (-1,2) = \big((-1,-1),(-1,2),(2,-1),(2,2)\big)[/itex]

but then part of me wonders if I've got it all wrong and it should really just be [itex]f((-1,2)) = ((1,4))[/itex] ..??

And then for part b:

b) [itex]f((-1,2]) = ... [/itex] ...

I don't really understand this at all since it has a square bracket which I'm led to believe means it represents a continuous interval of numbers not including that which is on the side of the curled bracket (according to this - interval notation). If that's the case, I don't know how to perform [itex]X \times Y[/itex] in the way I defined above.

And then we have stuff to do with [itex]f^{-1}[/itex] which is a whole other thing entirely.

(Just to check - am I right in saying that [itex]f^{-1}[/itex] on a set [itex]Y[/itex] is all the elements [itex]x \in X[/itex] such that [itex]f(x) \in Y[/itex] ??)

Or in other words: [itex]f^{-1}(Y) = \{ x \in X : f(x) \in Y \}[/itex] - right?

Hints much appreciated, thanks.