# Omega

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 Dec5 comment Advice for proving with induction scenarios with multiple chances for using the hypothesis. Thanks. Yeah, I know I asked for this scenario - this was out of curiosity. Dec5 comment Advice for proving with induction scenarios with multiple chances for using the hypothesis. Hm, I'm not very familiar with that, but it sounds interesting - could you elaborate a bit more? Dec5 comment Advice for proving with induction scenarios with multiple chances for using the hypothesis. If you were in my situation, looking at these four options, you'd also pick one by one? Well, I guess that's a relief, kinda lol. Dec5 comment Advice for proving with induction scenarios with multiple chances for using the hypothesis. @HagenvonEitzen: Yes, to choose which of the four terms to replace as my next step. Dec5 comment Proving $\frac{(n+1)^4}{4}+(n+1)^3\le\frac{(n+2)^4}{4}$ for all $n \ge 1$. @user112167: It actually came from an induction exercise. I added it there now. Dec4 comment Proving by induction inequalities that lack the variable on the right side. @SammyBlack: Someone told me that there is no problem with starting from $\frac{1}{n+1}$, because it just means that the succession begins from $\frac{1}{n+1}$ and ends on $\frac{1}{(n+1)+1}$. Is that right? Dec3 comment Proving by induction inequalities that lack the variable on the right side. @PraphullaKoushik: Well, it's an induction exercise, so I should use it. I don't recognise that term for inequality (or perhaps I do, but in Spanish) - I will take a look at it, thanks. Dec3 comment Proving by induction that $1+\frac{1}{2}+\frac{1}{3}+…+\frac{1}{n}\le\frac{n}{2}+1$ holds for all $n \ge 1$ @TenaliRaman: I know what they did, I don't know why, of all options, they had to choose that one. They could've added something else to both sides too, right? Why not? Nov9 comment Proving that $f(n,m) = 3^n5^m$ is injective. @AymanHourieh: Of couurseeee.. No, I fear not. I will check it out, thank you. Nov9 comment If $f$ is injective, demonstrate that $f \circ g = f \circ h \implies g = h$ Wait, what is the contradiction? Plus, in what part did you apply the fact that $f$ is injective? Nov9 comment How to work with function properties when it is defined by an equation system? I see. Thanks! That should be an answer. Nov8 comment Functions exercise for $f(x) = \begin{cases} x \textrm{ if } x \le 3 \\ 11 - 2x \textrm{ if } 3 < x\end{cases}$ Oh snap. Thanks for the observations! Nov8 comment How to determine the equivalence classes of a relation? Thanks! Why $\phi^{-1}(\{k\})$ and not $\phi^{-1}(k)$ by the way? Nov8 comment How to determine the equivalence classes of a relation? Could you elaborate more on what happened at "$\phi^{-1} ( \{ k \} )$ for $k = 0,1,2$"? I'm a little bit lost there. Nov7 comment Determining equivalence classes of certain pairs for the relation $(a,b)R(c,d) \iff a^2 + 7b^2 = c^2 +7d^2$ Just to be clear: if $d = 2$, then $7d^2 = 28$, so we would need $c^2 = -20$. However, we can't have a $c$ that fulfils $c^2 = -20$ right? So I guess it's not possible to have $d > 1$? Nov7 comment Determining the properties for the relation over $P(\mathbb{N})$ where $ARB \iff A \cup B \in H$ $A = \{1,2,3\}$ and $B = \mathbb{N}$, $A$ is not reflexive, $ARB$ holds since $\overline{B}$ is finite - but I still don't see how does that serve as a counterexample for transitivity. Nov7 comment Finding the first, last, minimal and maximal elements in these relations. @dfeuer: Just to be clear, what is your definition of "a set's class"? Nov7 comment Finding the first, last, minimal and maximal elements in these relations. @dfeuer: Without actually knowing their contents? I'm afraid not....... Unless.... Maybe, wouldn't it be $\emptyset$ in this case? Since $\emptyset \subseteq whateverSet$? Nov6 comment Is my transitivity proof correct for the relation over $\mathbb{Z} \times \mathbb{Z}$ where $(a,b)R(c,d) \iff (a \le c \lor b \le d)$? If a relation used single elements, would a metric line be a good analogy...? Nov6 comment Is my transitivity proof correct for the relation over $\mathbb{Z} \times \mathbb{Z}$ where $(a,b)R(c,d) \iff (a \le c \lor b \le d)$? Daaaaamn. Back to scratch. I really like the graph idea, I wonder if I could use that more often for the other relations I tried.