Esercizio 3. È data l'applicazione lineare \( F: \mathbb{R}_{\leq 2}[x] \rightarrow \mathcal{M}_{2}(\mathbb{R}) \) tale che \[ F(p(x))=\left(\begin{array}{cc}p(1) & p(0) \\ p(0) & p(-1)\end{array}\right) \] (a) Senza effettuare calcoli, dire perché esiste almeno un vettore di \( \mathcal{M}_{2}(\mathbb{R}) \) con controimmagine vuota sotto l'azione di \( F \). (b) Si calcoli la matrice associata ad \( F \) rispetto alla base \( \mathcal{B}=\left\{x, x+1, x^{2}-1\right\} \) di \( \mathbb{R}_{\leq 2}[x] \) e alla

12. Given set \( \mathrm{A}=\{\emptyset,\{\emptyset\}, 0\} \), which of the following is true? C. \( \emptyset \subseteq A \) B. \( \{\emptyset\} \in A \) C. \( \{\{\emptyset\}\} \subseteq A \) D. All are correct 13. For any sets \( A \) and \( B \) which of the following is true. A. If \( B \subseteq A \), then \( B^{\prime} \subseteq A^{\prime} \) C. \( A-B=A \cap B^{\prime} \) B. If \( A \subseteq B \), then \( A \cup B=B \) D. all except A. Part Three: solve the following problems. 1. For any propositions \( p \) and \( q \), determine whether \( (p \wedge q) \Rightarrow(q \wedge \neg p) \) is tautology, contradiction or neither? 2. Show that the argument: \( p \Rightarrow q, \neg p \Rightarrow h, h \Rightarrow k \vdash \neg q \Rightarrow k \) is valid. 3. Let \( \mathrm{U}= \) the set of one digit counting numbers \( A=\{x: x \) is an even number in \( U\} \) \( B=\{x: x \) is a square number in \( U\} \) \( \mathrm{C}=\{x: x+2<10 \), where \( x \in U\} \), then find the following. A. \( (A \cup B) / C \) C. \( \left(A \cap B^{\prime}\right) \Delta C \) B. \( (A \Delta B) / C \) D. \( \left(A \cap B^{\prime}\right)^{\prime} \) 4. Write the converse, inverse and contrapositive of the conditional statement. "If it is cold or rainy, I will not take a walk."

3) What type of graph should be made for the experiment in Model 1? * 1 point Scatterplot Bar graph Pie chart 4) What is the relationship between the number of paper clips and distance * 1 point that the paper airplane traveled in Model 1? As the number of paperclips increases, the distance decreases. As the number of paperclips decreases, the distance increases. As the number of paperclips increases, the distance increases.

Let \( A \) and \( B \) be two square matrices of order 2 such that \( A^{-1}=\left(\begin{array}{ll}1 & 1 \\ 1 & 2\end{array}\right) \) and \( B^{-1}=\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right) \). Then find the inverses of the matrices \( A^{-1}, A B \) and \( A^{t} \).

Partie A: Évaluation des ressources (15,5points) Exercice 1: ( 3,5 points) Le plan complexe \( P \) est muni d'un repère orthonormé direct \( (O ; \vec{u} ; \vec{v}) \). A et \( B \) sont des points du plan d'affixes respectives \( i \) et \( -2+i . F \) est l'application de \( P-\{A\} \) vers \( P \) qui, à tout point \( M(z) \) associe le point \( M^{\prime}\left(z^{\prime}\right) \) tel que \( z^{\prime}=i\left(\frac{z+2-i}{z-i}\right) \). Pour tout entier naturel \( n \), on désigne par \( C_{n} \) le point d'affixe \( (\sqrt{3}-i)^{n} \). On considère dans \( \mathbb{C} \) l'équation \( (E): z^{2} \cos ^{2} \alpha-2 z \sin 2 \alpha+1=0 \) où \( \left.\alpha \in\right] 0 ; \frac{\pi}{2}[ \). 1) Pour quelles valeurs de \( n \) les points \( C_{n} \) appartiennent-ils à l'axe des réels ? 2) Déterminent \( n \) pour que les points \( C_{n} \) appartiennent à la première bissectrice. \( 0,75 p t \) 3) Déterminer l'ensemble des points invariants par \( F \). 0,5pt 4) Déterminer les parties réelle et imaginaire de \( z^{\prime} \) 0,5pt 5) Résoudre dans \( \mathbb{C} \), l'équation \( (E) \). \( 1 p t \) 0,75pt

¿Qué es un cuantificador en lógica matemática? a) Un número que representa una cantidad fija b) Un simbolo que indica cuántos elementos de un conjunto cumplen una propiedad c) Un tipo de operacion matemática d) Una variable algebraica