Exercises

Exercises 4.3 Exercises

Use truth tables to determine whether or not the following pairs of statements are logically equivalent.

🔗
“ $(\sim P) \lor Q$ ” and “ $P \Rightarrow Q$ ”.
🔗
“ $P \Leftrightarrow Q$ ” and “ $(\sim P) \Leftrightarrow (\sim Q)$ ”.
🔗
“ $P \Rightarrow (Q \lor R)$ ” and “ $P \Rightarrow ((\sim Q) \Rightarrow R)$ ”.
🔗
“ $(P \lor Q) \Rightarrow R$ ” and “ $(P \Rightarrow R) \land (Q \Rightarrow R)$ ”.
🔗
“ $P \Rightarrow (Q \lor R)$ ” and “ $(Q \land R) \Rightarrow P$ ”.

Use the logical equivalences given in Theorem 4.2.3 and Theorem 4.2.7 to negate the following sentences.

🔗
$8$ is even and $5$ is prime.
🔗
If $n$ is a multiple of $4$ and $6,$ then it is a multiple of $24 .$
🔗
If $n$ is not a multiple of $10,$ then it is a multiple of $2$ but is not a multiple of $5 .$
🔗
$3 \leq x \leq 6 .$
🔗
A real number $x$ is less than $- 2$ or greater than $2$ if its square is greater than $4 .$
🔗
If a function $f$ is differentiable everywhere then whenever $x \in R$ is a local maximum of $f$ we have $f^{'} (x) = 0 .$

Show that the following pairs of statements are logically equivalent using Theorem 4.2.3.