Notation: $\overline{p}$ (not $p$), $p \land q$ ($p$ and $q$), $p \vee q$ ($p$ or $q$), $p \rightarrow q$ (if $p$, then $q$), $p \leftrightarrow q$ ($p$ if and only if $q$).
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The statements below are those from ex. 3 of last lesson’s worksheet (plus a new one). Denote the simple propositions by $p$ and $q$, write the statement symbolically and now determine the truth value of the whole statement.
Statement Symbolic form Truth value a) “The number $12$ is divisible by $3$ and by $4$.” b) “The number $15$ is even or is a multiple of $5$.” c) “If $2 > 3$, then $2^{2} > 3^{2}$.” d) “$\sqrt{5}$ is not a rational number.” e) “$- 3 < 2$ if and only if $( - 3)^{2} < 2^{2}$.” -
Consider the propositions $p$: “$\sqrt{49} = 7$” and $q$: “$2^{3} = 6$”. Give $v(p)$ and $v(q)$, then complete the table.
$v(p)$ = … ; $v(q)$ = …
Complete the table.
$\overline{p}$ $\overline{q}$ $p \land q$ $p \vee q$ $p \rightarrow q$ $q \rightarrow p$ $p \leftrightarrow q$ $\overline{p} \vee \overline{q}$ -
Write the negation of each proposition and give the truth value of the proposition and of its negation.
Proposition $p$ $v(p)$ Negation $\overline{p}$ $v(\overline{p})$ a) “The number $51$ is prime.” b) “$\sqrt{2} > 1,5$” c) “$\lbrack - 2,5\rbrack = - 3$” d) “$\left| 1 - \sqrt{3} \right| = 1 - \sqrt{3}$” -
Complete the truth table. Compare the last three columns. What do you notice?
Complete the table.
$p$ $q$ $\overline{p}$ $\overline{q}$ $\overline{p} \vee q$ $p \rightarrow q$ $p \land \overline{q}$ $1$ $1$ $1$ $0$ $0$ $1$ $0$ $0$ Observation: …
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Knowing that the proposition $p \rightarrow q$ is false, find $v(p)$ and $v(q)$, then the truth value of the propositions $p \vee q$, $\overline{p} \leftrightarrow q$ and $q \rightarrow p$.
a) $v(p)$ = … ; $v(q)$ = …
b) $v(p \vee q)$ = …
c) $v(\overline{p} \leftrightarrow q)$ = …
d) $v(q \rightarrow p)$ = …
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Consider the predicates $p(x)$: “$x^{2} = 4$” and $q(x)$: “$x > 0$”, $x \in \left\{ - 2,\ - 1,\ 0,\ 1,\ 2 \right\}$. Find the values of $x$ for which the proposition is true:
a) $p(x) \land q(x)$: $x \in$ …
b) $p(x) \vee q(x)$: $x \in$ …
c) $p(x) \rightarrow q(x)$: $x \in$ …