Uestion vo 3\( \} \) Examine whether the following set of vectors is a Basis and if so fint imensions of the Vector Space \( \mathrm{B}=\left\{\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right]\right\} \) of Vector Space of all \( 2 \times 2 \) diagonal matrices

¿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

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

Part II. The following problems have exactly one best answer. Chose the correct answer. 1. Which of the following compound propositions is necessarily true? A. \( p \wedge \neg p \) C. \( p \Leftrightarrow \neg p \) B. \( p \vee \neg p \) D. . \( (p \vee \neg p) \Rightarrow p \) 2. Which of the following represents a set? A. The Collection of good students in a class. C. \( \{a, b\}, c \). B. The collection of cows with 6 legs. D. The collection of interesting birds. 3. Which of the following is a proposition? A. What is your name? C. May God bless you! B. \( 2 x-3=0 \) D. 123 is a prime number. 4. Let A be any set and \( \mathrm{P}(\mathrm{A}) \) be the power set of A , which of the following is true? A. \( \{\varnothing\} \subseteq A \) B. \( \emptyset \in A \) C. \( \emptyset \in P(A) \) D. \( A \subseteq P(A) \) 5. Which of the following statement is true? A. \( (\forall x \in R)(\exists y \in R)(2 x+y=-4) \) C. \( (\forall x \in R)(\exists y \in R)(x y=1) \) B. \( (\exists x \in R)(\exists y \in R)(|x y|=-3) \) D. \( (\exists x \in R)(\forall y \in R)(2 x+y=-4 \) 6. Let A be any set with certain number of elements, then which of the following is false? A. A may contain 5 proper subsets C. A may contain 16 subsets B. A may contain only one proper subset D. A may contain only one subset 7. Let p and q be propositions which of the following implies that. \( p \wedge q \) is true. A. . \( p \vee q \) is true B. . \( \neg p \wedge q \) is false C. \( p \Rightarrow q \) is true D. \( p \Rightarrow \neg q \) is false

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.

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."