Esercizio 13.16 Costruire, se possibile, un'applicazione lineare \( F: \mathbb{R}^{3} \longrightarrow \mathbb{R}^{4} \) abbia come nucleo il sottospazio \( U=\mathcal{L}((1,1,0),(-1,1,1)) \). È unica tale applicazionet Svolgimento Deve essere \[ F(1,1,0)=F(-1,1,1)=(0,0,0,0) \] Poiché manca un vettore, per completare ad una base dello spazio di partenza, le applicazioni lineari possibili sono infinite. Si badi a non far sì che anche il terros vettore della base di partenza vada a finire nel vettore nullo, altrimenti si ottiene l'applicazione nulla che ha nucleo di dimensione tre. Lavorando un po' con l'occhio allenato si trova ad esempio \[ F(x, y, z)=(x-y+2 z, 0,0,0) \]

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. . \( (p \vee \neg p) \Rightarrow \mathrm{p} \) 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."

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) \( \quad p \vee q \leftrightarrow(r \wedge q) \).

Part B Eating daily from all the food groups will give you a balanced diet. Name the basic food groups.