Solve the problems in your notebook, with justifications.
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Build the truth table and decide which of the formulas are tautologies:
a) $p \rightarrow (p \vee q)$;
b) $(p \vee q) \rightarrow p$;
c) $\overline{p \land \overline{p}}$;
d) $(p \rightarrow q) \leftrightarrow (\overline{q} \rightarrow \overline{p})$.
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Prove the following equivalences with truth tables:
a) $\overline{p \land q} \equiv \overline{p} \vee \overline{q}$ (De Morgan's law);
b) $p \rightarrow (q \rightarrow r) \equiv (p \land q) \rightarrow r$.
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Write the negation of each proposition in words. For a)–d), state the truth value of the proposition and of its negation.
a) “The number $35$ is divisible by $5$ and by $7$.”
b) “$\sqrt{9} = 3$ or $\sqrt{9} = - 3$.”
c) “If $10 \vdots 4$, then $10 \vdots 8$.”
d) “$2 < \sqrt{5} < 3$” (that is, “$2 < \sqrt{5}$ and $\sqrt{5} < 3$”).
e) “Triangle $ABC$ is isosceles or right-angled.”
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Let $x$ be a real number. Write the negation of each statement:
a) “$x > 1$ and $x \leq 5$”;
b) “$x < - 2$ or $x \geq 3$”;
c) “If $x^{2} = 9$, then $x = 3$.”
For c), find a value of $x$ for which the negation is true.
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Write the converse, the inverse and the contrapositive of each implication, and state the truth value of the four statements:
a) “If a quadrilateral is a square, then its diagonals are congruent.”
b) “If a natural number is even, then its square is even.”
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Given that the proposition $(p \vee q) \rightarrow r$ is false and that $v(p) = 0$, find $v(q)$, $v(r)$ and the truth value of the proposition $\overline{q} \rightarrow (p \land r)$.
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Let $p$: “Ana goes to the cinema” and $q$: “Ana studies”.
a) Express in words: $\overline{p} \land q$, $\overline{p \vee q}$, $p \rightarrow \overline{q}$, $\overline{p \rightarrow q}$.
b) Which of these propositions says the same thing as “Ana neither goes to the cinema nor studies”?
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How many rows does the truth table of a formula with $3$ variables have?
And with $n$ variables?
Build the table of the formula $(p \vee q) \rightarrow r$ and state for which values of $p$, $q$, $r$ the formula is false.
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Challenge. One of Radu, Sorin and Tudor ate the cake. Radu says: “Sorin ate it.” Sorin says: “I did not eat it.” Tudor says: “I did not eat it.” Knowing that exactly one of them tells the truth, find out who ate the cake. (Hint: compare what Radu and Sorin say.)