The language has three unary function symbols f,g and h. The domain is a square if the model is a model of this sentence and the domain is finite.
(c) Let $\mathcal{L}^*\subseteq\mathcal{L}^{\text{FOPC}}$ be a first order language of predicate calculus consisting of three unary function symbols $f,g,h$.  (i) Write down an $\mathcal{L}^*$-sentence $\phi$ such if $\mathcal{A}$ is an $\mathcal{L}^*$-structure with $\mathcal{A}\vDash\phi$ then the restriction of $h_\mathcal{A}$ to the range of $f_\mathcal{A}$ gives a one-to-one function onto the range of $g_\mathcal{A}$.  (ii) Write down an $\mathcal{L}^*$-sentence $\psi$ such that if $\mathcal{A}$ is an $\mathcal{L}^*$-structure with $\mathcal{A}\vDash\psi$ and the domain $A$ of $\mathcal{A}$ is finite then the cardinality $|A|$ is a square.    Part (i) is easy enough, but I'm having trouble with part (ii). I assume that you can use part (i). My initial idea was to use $h$ to ensure $|Range(f_A)| =|Range(g_A)|$ but I'm not sure where to go from there. Thanks.
How to change symbol for footnote in minipage \renewcommand method does not work  \documentclass{article} \renewcommand{\thefootnote}{\arabic{footnote}} \begin{document}  \begin{minipage}{1.3\textwidth}  test this\footnote{abc} \end{minipage}  \end{document}   In the following example the footnote number is incorrect. Why?  \documentclass{article} \renewcommand{\thempfootnote}{\arabic{mpfootnote}} \begin{document} \begin{minipage}{.5\textwidth} test this\footnotemark \footnotetext{abc} \end{minipage} \end{document} How can I make a camera the active one? I have multiple cameras in a scene. What I want to do is to change which one is active. This way I can view the scene from different angles. How would I do this? How to change IP address to point to localhost? Redefining "cited on" string (and others) in biblatex I'm trying to modify the strings biblatex uses to set the "cited on" string used in the bibliography to identify the pages where a given references has been cited.    What I gather from the biblatex documentation (e.g., § 4.9.2.16) is that something like   \renewbibmacro{backrefpage}{&lt;newtext&gt;} \renewbibmacro{backrefpages}{&lt;newtext&gt;}   ought to work, but they have no effect for me (I'd also like to get this working for the "visited on" string, urlseen).  My MWE is unchanged from , with the addition of these (and related lines) just about anywhere. Quite simply (and similarly to my other questions about previous versions of this feat  and ) I want to know if this feat is balanced and if not then what should be done to improve it? Since the last iteration I have changed the wording as per David Coffron's answer, increased the time and monetary cost as well as added in the action requirement to activate the rune as per Ruse's answer and increased the costs as per linksassin's answer     Rune Caster      Prerequisites: ability to cast at least one spell of 1st level or higher      When you take this feat you can imbue your spells of 1st level or higher into a rune by casting the spell as a rune, this means that you must have the object you want to put the rune on within reach for the entire duration of the casting as well as any tools required to make the rune.      When casting a rune you add the time listed in the cost time table to the casting time and the cost listed in the cost time table to the components of the spell in accordance with the stored spell.      \begin{array}{|l|l|l|} \hline \text{Spell level} &amp; \text{Time (hrs)} &amp; \text{Cost (GP)}\\ \hline 1 &amp; 1 &amp; 50 \\ \hline 2 &amp; 2 &amp; 100\\ \hline 3 &amp; 4 &amp; 500\\ \hline 4 &amp; 6 &amp; 1,000\\ \hline 5 &amp; 8 &amp; 5,000\\ \hline 6 &amp; 12 &amp; 10,000\\ \hline 7 &amp; 16 &amp; 50,000\\ \hline 8 &amp; 18 &amp; 100,000\\ \hline 9 &amp; 24 &amp; 500,000\\ \hline \end{array}      When making a rune you choose a trigger which is touch and/or command word and can be specified to only a group of people/creatures or a body part. When defining creatures that can activate the rune you may be as specific or general as you want. The DM interprets the designated creatures as your character would have interpreted them when you made the rune.      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I must prove the following using mathematical induction:     For all $n\in\mathbb{Z^+}$, $1+2+2^2+2^3+\cdots+2^n=2^{n+1}-1$.   This is what I have in my proof so far:     Proof: Let $p\left(n\right)$ be $\sum_{j=0}^n 2^j=1+2+2^2+\cdots+2^n=2^{n+1}-1$. Then $P\left(0\right)$ is    \begin{align} \sum\limits_{j=0}^0 2^j=1=2^1-1.\tag{1} \end{align} Hence    $P\left(0\right)$ is true. Next, for some $k\in\mathbb{Z^+}$, assume    $P\left(k\right)$ is true. Then \begin{align}  \sum\limits_{j=0}^{k+1}2^j&amp;=\left(\sum\limits_{j=0}^k  2^j\right)+2^{k+1}\tag{2}\\ &amp;=2^{k+1}-1+2^{k+1}\tag{3} \\ &amp;=2\cdot 2^{k+1}-1\tag{4}\\ &amp;=2^{k+2}-1\tag{5}\\ &amp;=2^{\left(k+1\right)+1}-1.\tag{6} \end{align}   Therefore, $P\left(n\right)$ holds for all $n$.$\;\;\;\;\blacksquare$   Is everything right? My professor went over the methodology in class quite fast (and I had a bit too much coffee that night) and I want to make sure I have it down for tomorrow night's exam. Let $A,B$ be two sets. 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