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MATERI III PROPOSISI

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2 Rules of Inference

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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

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1.M CGiven 2. D CGiven 3.D SGiven 4. MGiven 5.C Elimination (1,4) 6.C DContrapositive of 2 7.C STransitivity (6,3) 8.SModus Ponens (5,7) Then, we conclude that Ellen is smart. Proofs - A little proof…

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ALJABAR PROPOSISI Idempotenp v p ≡ p p ᴧ p ≡ p Asosiatif (p ᴧ q) ᴧ r ≡ p ᴧ (q ᴧ r) (p v q) v r ≡ p v (q v r) Komutatif p v q ≡ q v p p ᴧ q ≡ q ᴧ p Distributifp ᴧ (q v r) ≡ (p ᴧ q) v (p ᴧ r) p v (q ᴧ r) ≡ (p v q) ᴧ (p v r)

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ALJABAR PROPOSISI Identitas p v f ≡ p p v t ≡ t p ᴧ f ≡ f p ᴧ t ≡ p Komplemen ̴t ≡ f̴f ≡ t p v ̴p ≡ t p ᴧ ̴p ≡ f Involution ̴( ̴p) ≡ p De Morgan’s ̴(p ᴧ q) ≡ ̴p v ̴q ̴(p v q) ≡ ̴p ᴧ ̴q

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ALJABAR PROPOSISI Absorpsip v (p ᴧ q) ≡ p p ᴧ (p v q) ≡ p Implikasip → q ≡ ̴p v q Biimplikasip ↔ q ≡ (p → q)ᴧ(q → p) Kontraposisip → q ≡ ̴ q → ̴ p

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

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Exercises Use truth tables to determine whether the following argument forms are valid. -Jika sistem digital maka akurat dan jika gerbang logika maka aljabar Boole. -Sistem digital atau gerbang logika -Tidak akurat atau bukan aljabar Boole -Akurat Δ Sistem digital 9

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

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