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CS70:Proofs Today!!!

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CS70:Proofs Today!!! Woohoo!!
Propositional Logic Identities. ¬ ∀ x P ( x ) ≡ ∃ x ¬ P ( x )

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Propositional Logic Identities. ¬ ∀ x P ( x ) ≡ ∃ x ¬ P ( x ) ¬ ∃ x P ( x ) ≡ ∀ x ¬ P ( x )
Propositional Logic Identities. ¬ ∀ x P ( x ) ≡ ∃ x ¬ P ( x ) ¬ ∃ x P ( x ) ≡ ∀ x ¬ P ( x )

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Propositional Logic Identities. ¬ ∀ x P ( x ) ≡ ∃ x ¬ P ( x ) ¬ ∃ x P ( x ) ≡ ∀ x ¬ P ( x ) ¬ ( P Q ) ≡ ¬ P ∨¬ Q
Propositional Logic Identities. ¬ ∀ x P ( x ) ≡ ∃ x ¬ P ( x ) ¬ ∃ x P ( x ) ≡ ∀ x ¬ P ( x ) ¬ ( P Q ) ≡ ¬ P ∨¬ Q ¬ ( P Q ) ≡ ¬ P ∨¬ Q

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Propositional Logic Identities. ¬ ∀ x P ( x ) ≡ ∃ x ¬ P ( x ) ¬ ∃ x P ( x ) ≡ ∀ x ¬ P ( x ) ¬ ( P Q ) ≡ ¬ P ∨¬ Q ¬ ( P Q ) ≡ ¬ P ∨¬ Q
Propositional Logic Identities. ¬ ∀ x P ( x ) ≡ ∃ x ¬ P ( x ) ¬ ∃ x P ( x ) ≡ ∀ x ¬ P ( x ) ¬ ( P Q ) ≡ ¬ P ∨¬ Q ¬ ( P Q ) ≡ ¬ P ∨¬ Q P = Q ≡ ¬ P = ⇒ ¬ Q contrapositive.

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Propositional Logic Identities. ¬ ∀ x P ( x ) ≡ ∃ x ¬ P ( x ) ¬ ∃ x P ( x ) ≡ ∀ x ¬ P ( x ) ¬ ( P Q ) ≡ ¬ P ∨¬ Q ¬ ( P Q ) ≡ ¬ P ∨¬ Q P = Q ≡ ¬ P = ⇒ ¬ Q contrapositive. P = Q ≡ ¬ P Q
Propositional Logic Identities. ¬ ∀ x P ( x ) ≡ ∃ x ¬ P ( x ) ¬ ∃ x P ( x ) ≡ ∀ x ¬ P ( x ) ¬ ( P Q ) ≡ ¬ P ∨¬ Q ¬ ( P Q ) ≡ ¬ P ∨¬ Q P = Q ≡ ¬ P = ⇒ ¬ Q contrapositive. P = Q ≡ ¬ P Q P = Q

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Propositional Logic Identities. ¬ ∀ x P ( x ) ≡ ∃ x ¬ P ( x ) ¬ ∃ x P ( x ) ≡ ∀ x ¬ P ( x ) ¬ ( P Q ) ≡ ¬ P ∨¬ Q ¬ ( P Q ) ≡ ¬ P ∨¬ Q P = Q ≡ ¬ P = ⇒ ¬ Q contrapositive. P = Q ≡ ¬ P Q P = Q P is antecedent or assumption.
Propositional Logic Identities. ¬ ∀ x P ( x ) ≡ ∃ x ¬ P ( x ) ¬ ∃ x P ( x ) ≡ ∀ x ¬ P ( x ) ¬ ( P Q ) ≡ ¬ P ∨¬ Q ¬ ( P Q ) ≡ ¬ P ∨¬ Q P = Q ≡ ¬ P = ⇒ ¬ Q contrapositive. P = Q ≡ ¬ P Q P = Q P is antecedent or assumption. Q is consequent or conclusion.

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Review puzzle Theory: If you drink alcohol you must be at least 18.
Review puzzle Theory: If you drink alcohol you must be at least 18. Which cards do you turn over?

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Review puzzle Theory: If you drink alcohol you must be at least 18. Which cards do you turn over? Write implication and contraposition:
Review puzzle Theory: If you drink alcohol you must be at least 18. Which cards do you turn over? Write implication and contraposition: Drink = 18”

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Review puzzle Theory: If you drink alcohol you must be at least 18. Which cards do you turn over? Write implication and contraposition: Drink = 18” < 18” = Don’t Drink.
Review puzzle Theory: If you drink alcohol you must be at least 18. Which cards do you turn over? Write implication and contraposition: Drink = 18” < 18” = Don’t Drink. Its easier now.

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Review puzzle Theory: If you drink alcohol you must be at least 18.
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• Fall '08
• Rappaport
• Natural number, Prime number, Divisor, contraposition, natural number p

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