3 Consider \( U=\left\{x \mid x \leqslant 12, x \in \mathbb{Z}^{+}\right\} \), \[ A=\{2,7,9,10,11\} \text { and } B=\{1,2,9,11,12\} \text {. } \] a Show these sets on a Venn diagram. \( \begin{array}{llll}\text { b List the elements of: } & \text { I } A \cap B & \text { II } A \cup B & \text { III } B^{\prime} \\ \text { c Find: } 1 n(A) & \text { II } n\left(B^{\prime}\right) & \text { III } n(A \cap B) & \text { iv } n(A \cup B)\end{array} \)

2) a) Determine the whether the following expression is tautology, contradiction or contingency. i) \( \quad p \rightarrow((q \vee r) \wedge(p \wedge q)) \). ii) \( \quad p \vee q \leftrightarrow(r \wedge q) \).

Soit A une matrice carrée d'ordre \( N \) inversible, \( b, c, f \in \mathbb{R}^{N} \). Soient \( \alpha, \gamma \in \mathbb{R} \). On cherche à résoudre le système suivant avec \( x \in \mathbb{R}^{N}, \lambda \in \mathbb{R} \) \[ \left\{\begin{array}{l}A x+b \lambda=f \\ c x+\alpha \lambda=\gamma\end{array}\right. \] 1) Ecrire le système ci-dessus sous la forme :My=g, avec \( M \) matrice carrée \( N+1, y \in \mathbb{R}^{N+1}, g \in \mathbb{R}^{N+1} \) Donner l'expression de \( M, y \) et \( g \) 2) Donner une relation entre A,b,c et \( \alpha \) qui soit une condition nécessaire et sL pour que le système ci-dessus soit inversible. Dans toute la suite on supposera, que cette relation est vérifiée.

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

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."

11 If \( \overrightarrow{\mathrm{A}}=(4,3,2) \quad, \overrightarrow{\mathrm{B}}=(-2,5,-1), \overrightarrow{\mathrm{C}}=(8,19,4) \) (1) Find in terms of fundamental unit vectors each of : \( \vec{B}-\overrightarrow{\mathrm{C}}, 2(\overrightarrow{\mathrm{~B}}+\overrightarrow{\mathrm{C}}) \) (2) Find : \( \overrightarrow{\mathrm{C}} \) in terms of \( \overrightarrow{\mathrm{A}} \) and \( \overrightarrow{\mathrm{B}} \)