I was given problems to do over the summer in preparation for my graduate course study; however, it has been about 2 years since my discrete mathematics days and I am really stumbling on a couple problems. Below I listed two problems that are giving me problems. I am not really searching for answers, but explanations of how to do each one because I know how important these concepts are.
- Let X := R. For A, B ⊂ R, we say A ≥ B if B ⊆ A. Is this relation complete, symmetric, transitive, and/or reflexive?
I know what each "thing" (complete, symmetric, transitive, reflexive) is and I could answer this with ease, but I don't know how to deal with an inequality between sets. A and B are both sets I believe and I don't know how to say set A would be greater than set B.
- Let f : R → R with f(x) := x^3, A := (−8, 1) and B := [−1, 8]. Find f(A) and the inverse of f(A).
This problem is giving me a problem because A and B are sets. It is extremely easy, when A is a given number. My best guess was to just plug in the -8 into the equation and the 1 into the equation for the inverse, but I don't think this is right.
Any help is appreciated. I know these problems a basic problems, but for some reason I just cannot remember the material. Thanks
For 1., we are simply declaring "$A \ge B$" to mean $B \subseteq A$, i.e. $B$ is a subset of $A$.
For 2., in general $f(A)$ denotes $\{f(x) : x \in A\}$, that is, the set of all possible values $f$ takes when you plug in things from $A$. This is known as the image of $f$.
I'm not sure what "inverse of $f(A)$" means.Likewise, $f^{-1}(A)$ denotes $\{x : f(x) \in A\}$, that is, all things in the domain ($\mathbb{R}$) that after plugging into $f$, gives something in $A$. This is the "inverse image" or preimage of $A$.