Disjunction reasoning rules: pencil-and-paper exercises (5 exercises)

First, attempt each exercise on your own, and then compare your answer with the worked-out solution.

Show that $p\lor q\vdash q\lor p$ .

Worked-out solution

disjunctie redeneerrefels opdracht 1

Of course, the derivation below is also correct.

disjunctie-redeneerregel opdracht 1

Show that $p\lor (q\lor r)\vdash (p\lor q)\lor r$ .

Worked-out solution

disjunctie-redeneerregels opdracht 2

Of course, the derivation below is also correct.

disjunctie-redeneerregels opdracht 2

Show that $p\land (q\lor r)\vdash (p\land q)\lor (p\land r)$ .

Worked-out solution

disjunctie-redeneerregels opdracht 3

Of course, the derivation below is also correct.

disjunctie-redeneerregels opdracht 3

Show that $(p\land q)\lor (p\land r)\vdash p\land (q\lor r)$ .

Worked-out solution

disjunctie-redeneerregels opdracht 4

Of course, the derivation below is also correct.

disjunctie-redeneerregels opdracht 4

Note that the results of exercises 3 and 4 provide evidence of the following tautology about distributivity of $\land$ and $\lor$ operators:

$p\land(q\lor r)\leftrightarrow (p\land q)\lor (p\land r)$

Show that $p\rightarrow q, r\rightarrow s, p\lor r \vdash q \lor s$ .

Worked-out solution

disjunctie-redeneerregels opdracht 5

Of course, the derivation below is also correct.

disjunctie-redeneerregels opdracht 5