contoh pengerjaan soal logika matematika

1.a. p ∨ q ∨ ( ¬p ∨ ¬q ∧ r )
⇔¬p ∧ ¬q ∧ (p ∧ q ∨ ¬r)
⇔¬p ∧ ¬q ∧ ((p ∧ q) ∨ ¬r)
⇔¬p ∧ ¬q ∧ (¬r ∨ p) ∧ (¬r ∨ p)
⇔¬p ∧ (p ∨ ¬r) ∧ ¬q ∧ (q ∨ ¬r)
⇔(¬p ∧ ¬r) ∧ (¬q ∧ ¬r)
⇔¬p ∧ ¬q ∧ ¬r ∧ ¬r
⇔¬p ∧ ¬q ∧ ¬r

b. p ∧ ( q ∨ r ) ∧ ( ¬p ∨ ¬q ∨ r )
⇔¬p ∨ (¬q ∧ ¬r) ∨ (p ∧ q ∧ ¬r)
⇔¬p ∨ (p ∧ q ∧ ¬r) ∨ (¬q ∧ ¬r)
⇔(¬p ∨ p) ∧ (¬p ∨ q) ∧ (¬p ∨ ¬r) ∨ (¬q ∧ ¬r)
⇔To ∧ (¬p ∨ q) ∧ (¬p ∨ ¬r) ∨ (¬q ∧ ¬r)
⇔(¬p ∨ q) ∧ (¬p ∨ ¬r) ∨ (¬q ∧ ¬r)
⇔¬p ∨ (q ∧ ¬r) ∨ (¬q ∧ ¬r)
⇔¬p ∨ ¬r ∧ (¬q ∧ ¬r)
⇔¬p ∨ ¬r ∧ To
⇔¬p ∨ ¬r


2.( p → q) ∧ [ ¬q ∧ ( r ∨ ¬q )]                                             Alasan

⇔ ( p → q) ∧ ¬q                                                                     De morgan’s

⇔ ( ¬p ∨ q ) ∧ ¬q                                                                  Assosiatif

⇔ ¬q ∧ ( ¬p ∨ q )                                                                  Assosiatif

⇔ ( ¬q ∧ ¬p ) ∨ ( ¬q ∧ q )                                                    Distributive

⇔ ( ¬q ∧ ¬p ) ∨ F0                                                                Inverse

⇔ ¬q ∧ ¬p                                                                              Identity

⇔ ¬( q ∨ p )                                                                            De morgan’s