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Index stranica OSTALE MATEMATIČKE OBLASTI MATEMATIČKA LOGIKA

Tautologije

[inlmath]\left[p\land\left(q\Rightarrow\lnot p\right)\right]\Leftrightarrow\lnot\left(p\Rightarrow q\right)[/inlmath]
  • +1

Tautologije

Postod primus » Ponedeljak, 24. Maj 2021, 06:31

Nekoliko tautologija iz propozicione logike. Možda će nekome dobro doći za vežbanje.

Tautologije

[dispmath]\mathbf{1.} \, \lnot(p \Longrightarrow q) \Longrightarrow (\lnot p \Longrightarrow q)[/dispmath][dispmath]\mathbf{2.}\, (((p \land (q \Longrightarrow p)) \Longrightarrow p) \land q) \lor \lnot q[/dispmath][dispmath]\mathbf{3.}\, ((\lnot p \Longrightarrow r) \land (\lnot q \Longrightarrow r)) \iff (\lnot(p \land q) \Longrightarrow r)[/dispmath][dispmath]\mathbf{4.}\, (\lnot p \land r) \lor (\lnot q \land r) \iff \lnot(r \Longrightarrow (p \land q))[/dispmath][dispmath]\mathbf{5.}\, ((\lnot p \lor r) \land (\lnot q \lor r)) \iff ((p \lor q) \Longrightarrow r)[/dispmath][dispmath]\mathbf{6.}\, \lnot(p \land (q \lor r)) \iff (p \Longrightarrow \lnot q) \land (p \Longrightarrow \lnot r)[/dispmath][dispmath]\mathbf{7.}\, ((p \Longrightarrow q) \lor r) \iff ((p \lor r) \Longrightarrow (q \lor r))[/dispmath][dispmath]\mathbf{8.}\, ((p \Longrightarrow q) \land r) \iff (\lnot(\lnot p \land r) \Longrightarrow (q \land r))[/dispmath][dispmath]\mathbf{9.}\, ((p \Longrightarrow r) \lor (q \Longrightarrow r)) \iff ((p \land q) \Longrightarrow r)[/dispmath][dispmath]\mathbf{10.}\, ((p \Longrightarrow r) \land (q \Longrightarrow r)) \Longrightarrow ((p \land q) \Longrightarrow r)[/dispmath][dispmath]\mathbf{11.}\, (p \land q) \Longrightarrow ((p \Longrightarrow r) \Longrightarrow (q \Longrightarrow r))[/dispmath][dispmath]\mathbf{12.}\, (p \lor q) \Longrightarrow (\lnot(\lnot p \Longrightarrow r) \Longrightarrow \lnot(q \Longrightarrow r))[/dispmath][dispmath]\mathbf{13.}\, (p \Longrightarrow q) \Longrightarrow ((p \land r) \Longrightarrow (q \land r))[/dispmath][dispmath]\mathbf{14.}\, (p \land q) \lor (p \land r) \Longrightarrow q \lor r[/dispmath][dispmath]\mathbf{15.}\, \lnot p \Longrightarrow (((\lnot p \lor (q \Longleftrightarrow r)) \lor (q \land r)) \lor r)[/dispmath][dispmath]\mathbf{16.}\, (p \land q) \lor ((\lnot p \lor (\lnot p \iff r)) \lor r)[/dispmath][dispmath]\mathbf{17.}\, (((p \Longrightarrow q) \iff q) \lor(\lnot q \land (\lnot r \Longrightarrow \lnot q))) \lor q[/dispmath][dispmath]\mathbf{18.}\, ((\lnot p \land(\lnot q \Longrightarrow \lnot p)) \lor p) \lor ((q \iff r) \Longrightarrow q)[/dispmath][dispmath]\mathbf{19.}\, (p \Longrightarrow q) \lor (p \iff ((\lnot r \lor q) \lor p))[/dispmath][dispmath]\mathbf{20.}\, ((\lnot (p \land q) \land (p \Longrightarrow q))\lor q)\lor (r \Longrightarrow (\lnot(p \Longrightarrow q) \lor p))[/dispmath][dispmath]\mathbf{21.}\, ((((p \lor \lnot q) \lor r) \lor \lnot r) \lor (((q \Longrightarrow r) \Longrightarrow p) \land q)) \lor p[/dispmath][dispmath]\mathbf{22.}\, (((\lnot p \Longrightarrow q) \land \lnot r) \lor \lnot p) \lor p[/dispmath][dispmath]\mathbf{23.}\, (((p \Longrightarrow (q \lor r)) \land q) \lor (((q \lor r) \lor q) \iff \lnot q)) \lor \lnot q[/dispmath][dispmath]\mathbf{24.}\, ((\lnot p \iff q) \lor r) \lor (r \Longrightarrow q)[/dispmath][dispmath]\mathbf{25.}\, ((p \lor (q \land ((\lnot p \iff r) \iff p))) \lor ((q \Longrightarrow r) \land \lnot p)) \lor r[/dispmath][dispmath]\mathbf{26.}\, ((p \Longrightarrow q) \lor p) \lor (\lnot r \land q)[/dispmath][dispmath]\mathbf{27.}\, (( \lnot p \Longrightarrow \lnot q) \land (q \lor r)) \lor (r \Longrightarrow r)[/dispmath][dispmath]\mathbf{28.}\, ( \lnot p \Longrightarrow q) \lor (r \iff (r \iff \lnot((p \Longrightarrow q) \iff p)))[/dispmath][dispmath]\mathbf{29.}\, (((p \Longrightarrow ( q \iff \lnot r)) \iff q) \Longrightarrow r) \lor \lnot r[/dispmath][dispmath]\mathbf{30.}\, (\lnot ((p \Longrightarrow \lnot p) \land p) \lor q) \lor ((r \Longrightarrow \lnot p) \land p)[/dispmath][dispmath]\mathbf{31.}\, ((\lnot p \land q) \lor r) \lor ((\lnot q \iff (\lnot r \iff q)) \Longrightarrow p)[/dispmath][dispmath]\mathbf{32.}\, p \Longrightarrow (q \lor (r \Longrightarrow p))[/dispmath][dispmath]\mathbf{33.}\, \lnot p \Longrightarrow ((q \land r) \lor \lnot p)[/dispmath][dispmath]\mathbf{34.}\, p \Longrightarrow ((q \iff r) \Longrightarrow p)[/dispmath][dispmath]\mathbf{35.}\, (p \Longrightarrow \lnot q) \lor ((q \Longrightarrow r) \Longrightarrow p)[/dispmath][dispmath]\mathbf{36.}\, (\lnot p \Longrightarrow \lnot q) \lor ( \lnot ((q \land r) \land p) \iff \lnot p)[/dispmath][dispmath]\mathbf{37.}\, (\lnot p \Longrightarrow (((p \land q) \iff q) \Longrightarrow r)) \lor (q \Longrightarrow r)[/dispmath][dispmath]\mathbf{38.}\, ((p \lor q) \lor (\lnot q \Longrightarrow r)) \lor \lnot q[/dispmath][dispmath]\mathbf{39.}\, (((((p \Longrightarrow q) \Longrightarrow p) \Longrightarrow q) \lor r) \lor p) \lor (q \land (p \iff \lnot q))[/dispmath][dispmath]\mathbf{40.}\, ((p \lor (q \land r)) \Longrightarrow p) \lor q[/dispmath][dispmath]\mathbf{41.}\, ((\lnot p \lor ((q \iff r) \land (((s \iff p) \Longrightarrow q)\lor r))) \lor r) \lor p[/dispmath][dispmath]\mathbf{42.}\, (((p \Longrightarrow q) \iff r) \lor ((\lnot p \iff s) \land q)) \lor (p \iff p)[/dispmath][dispmath]\mathbf{43.}\, (((\lnot p \iff q) \Longrightarrow p) \lor q) \lor (r \lor s)[/dispmath][dispmath]\mathbf{44.}\, (( p \lor (q \iff q)) \lor \lnot (r \lor (q \iff s))) \lor r[/dispmath][dispmath]\mathbf{45.}\, p \Longrightarrow (((\lnot q \iff r) \land \lnot s) \lor p)[/dispmath][dispmath]\mathbf{46.}\, (((((p \iff q) \lor r) \lor ((p \iff q) \land p)) \lor \lnot s) \lor (p \iff q)) \lor (r \iff \lnot s)[/dispmath][dispmath]\mathbf{47.}\, ((\lnot p \lor (( p \land ((q \iff r) \iff \lnot r)) \land s)) \lor (r \iff p)) \lor p[/dispmath][dispmath]\mathbf{48.}\, (p \iff (q \land (q \Longrightarrow p))) \lor (((\lnot r \iff q) \land q) \Longrightarrow s)[/dispmath][dispmath]\mathbf{49.}\, (((((p \Longrightarrow q) \Longrightarrow q) \iff r) \Longrightarrow s) \lor (((r \land s) \lor r) \iff s)) \lor r[/dispmath][dispmath]\mathbf{50.}\, (p \land q) \lor ( \lnot (( \lnot r \iff r) \iff p) \Longrightarrow ( q \Longrightarrow (( \lnot s \lor (\lnot r \iff r)) \Longrightarrow s)))[/dispmath]
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