Propositional Logic: Natural deduction
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.
Exercise 1 Show that \(p\lor q\vdash q\lor p\).
Worked-out solution
Of course, the derivation below is also correct.
Exercise 2 Show that \(p\lor (q\lor r)\vdash (p\lor q)\lor r\).
Worked-out solution
Of course, the derivation below is also correct.
Exercise 3 Show that \(p\land (q\lor r)\vdash (p\land q)\lor (p\land r)\).
Worked-out solution
Of course, the derivation below is also correct.
Exercise 4 Show that \((p\land q)\lor (p\land r)\vdash p\land (q\lor r) \).
Worked-out solution
Of course, the derivation below is also correct.
Note that the results of exercises 3 and 4 provide evidence of the following tautology about distributivity of \(\land\) and \(\lor\) operators: \[p\land(q\lor r)\leftrightarrow (p\land q)\lor (p\land r)\]
Exercise 5 Show that \(p\rightarrow q, r\rightarrow s, p\lor r \vdash q \lor s \).
Worked-out solution
Of course, the derivation below is also correct.
