Notation: $v(p) = 1$ if the proposition $p$ is true and $v(p) = 0$ if $p$ is false. Do the calculations in your notebook; fill in the tables on the worksheet.
-
Decide which of the following statements are propositions. For those that are propositions, give their truth value.
Statement Proposition? (YES / NO) Truth value a) “$\sqrt{16} = 4$” b) “$2^{3} + 1 = 3^{2}$” c) “Solve the equation $x + 3 = 7$.” d) “$x + 3 = 7$” e) “$- 2 \in {\mathbb{N}}$” f) “Every prime number is odd.” g) “How beautiful geometry is!” h) “$\sqrt{2} \in {\mathbb{Q}}$” -
Find the truth value of each proposition and justify it by calculation.
a) $p$: “$(\sqrt{3} - 1)(\sqrt{3} + 1) = 2$” $v(p)$ = … b) $q$: “$\left\lbrack - \frac{5}{2} \right\rbrack = - 2$” (integer part) $v(q)$ = … c) $r$: “$\frac{1}{2} + \frac{1}{3} = \frac{2}{5}$” $v(r)$ = … d) $s$: “$\left| 1 - \sqrt{2} \right| = \sqrt{2} - 1$” $v(s)$ = … -
Identify the simple propositions that make up each statement and give the truth value of each simple proposition (not of the whole statement).
Statement The simple propositions and their truth values 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.” -
Consider the predicate $p(x)$: “$x^{2} - 5x + 6 = 0$”, $x \in \left\{ 0,\ 1,\ 2,\ 3,\ 4 \right\}$.
a) Complete the table.
$x$ $0$ $1$ $2$ $3$ $4$ $x^{2} - 5x + 6$ $v(p(x))$ b) Write the truth set of the predicate: $A_{p}$ = …
-
Consider the predicate $P(n)$: “$2^{n} > n^{2}$”, $n \in {\mathbb{N}}$.
a) Complete the table.
$n$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $2^{n}$ $n^{2}$ $v(P(n))$ b) What do you notice?
-
Consider the predicate $p(x,\ y)$: “$x + 2y = 7$”, $x,\ y \in {\mathbb{N}}$.
a) Give $v(p(1,\ 3))$ and $v(p(3,\ 1))$.
b) Find all pairs $(x,\ y)$ of natural numbers for which $p(x,\ y)$ is a true proposition.