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MATERI III

MATERI III. PROPOSISI. Rules of Inference. Proofs - A little proof…. Here’s what you know: Ellen is a math major or a CS major. If Ellen does not like discrete math, she is not a CS major. If Ellen likes discrete math, she is smart. Ellen is not a math major.

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MATERI III

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  1. MATERI III PROPOSISI

  2. Rules of Inference

  3. Proofs - A little proof… Here’s what you know: Ellen is a math major or a CS major. If Ellen does not like discrete math, she is not a CS major. If Ellen likes discrete math, she is smart. Ellen is not a math major. Can you conclude Ellen is smart? M  C D  C D  S M

  4. Proofs - A little proof… • M  C Given • D  C Given • D  S Given • M Given • C Elimination (1,4) • C  D Contrapositive of 2 • C  S Transitivity (6,3) • S Modus Ponens (5,7) • Then, we conclude that Ellen is smart.

  5. ALJABAR PROPOSISI • Idempoten p v p ≡ p p ᴧ p ≡ p • Asosiatif (p ᴧ q) ᴧ r ≡ p ᴧ (q ᴧ r) (p v q) v r ≡ pv (q vr) • Komutatif p v q ≡ q v p p ᴧ q ≡ q ᴧ p • Distributif p ᴧ (qv r) ≡ (p ᴧ q) v (p ᴧ r) p v (q ᴧ r) ≡ (pv q) ᴧ (p vr)

  6. ALJABAR PROPOSISI • Identitas p v f ≡ p p v t ≡ t p ᴧ f ≡ f p ᴧ t ≡ p • Komplemen ̴t ≡ f ̴f ≡ t p v ̴p ≡ t p ᴧ ̴p ≡ f • Involution ̴( ̴p) ≡ p • De Morgan’s ̴(p ᴧ q) ≡ ̴p v ̴q ̴(pv q) ≡ ̴p ᴧ ̴q

  7. ALJABAR PROPOSISI • Absorpsi p v (p ᴧ q) ≡ p p ᴧ (pv q) ≡ p • Implikasi p → q ≡ ̴p v q • Biimplikasi p ↔ q ≡ (p → q)ᴧ(q → p) • Kontraposisi p → q ≡ ̴ q → ̴ p

  8. Exercises Use truth tables to determine whether the following argument forms are valid.

  9. Exercises Use truth tables to determine whether the following argument forms are valid. • Jikasistem digital makaakuratdanjikagerbanglogikamakaaljabar Boole. • Sistem digital ataugerbanglogika • Tidakakuratataubukanaljabar Boole • Akurat Δ Sistem digital

  10. Exercises Simplify using proposition algebra (p ᴧ̴ (pv ̴ q)) v p ᴧ (pv q) ((pv q) ᴧ̴ p) v ̴ (pv q) v ( ̴ p ᴧ q) ( ̴ p ᴧ (q → ̴ r)) v ((p v r) ↔ q )

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