Quiz 19

Question 1

Assume ∀x ∀y P(x,y)

Answer 1

Assumption

Question 2

∀y P(x,y)

Answer 2

Universal Instantiation

Question 3

P(x,x)

Answer 3

Universal Instantiation

Question 4

∀x P(x,x)

Answer 4

Universal Generalization

Question 5

∴ ∀x ∀y P(x,y) → ∀x P(x,x)

Answer 5

Derivation

Question 6

Assume ∀x ¬P(x)

Answer 6

Assumption

Question 7

Assume ∃x P(x)

Answer 7

Assumption

Question 8

P(x)

Answer 8

Existential Instantiation

Question 9

∀x ¬P(x)

Answer 9

Copy

Question 10

¬P(x)

Answer 10

Universal Instantiation

Question 11

P(x)

Answer 11

Copy

Question 12

P(x) ∧ ¬P(x)

Answer 12

Conjunction

Question 13

0…p ∧ ¬p ≡ 0…∴ ∃x P(x) → 0

Answer 13

Derivation

Question 14

¬∃x P(x)

Answer 14

Contradiction

Question 15

∴ ∀x ¬P(x) → ¬∃x P(x)

Answer 15

Derivation

Question 16

Assume ¬∃x P(x)

Answer 16

Assumption

Question 17

Let x be arbitrary…Comment….Assume P(x)

Answer 17

Assumption

Question 18

∃x P(x)

Answer 18

Existential Generalization

Question 19

¬∃x P(x)

Answer 19

Copy

Question 20

∃x P(x)..∧ ¬∃x P(x)

Answer 20

Conjunction

Question 21

0…p ∧ ¬p ≡ 0…∴ P(x) → 0

Answer 21

Derivation

Question 22

¬P(x)

Answer 22

Contradiction

Question 23

∀x ¬P(x)

Answer 23

Universal Generalization

Question 24

∴ ¬∃x P(x) → ∀x ¬P(x)

Answer 24

Derivation

Question 25

∀x ¬P(x) → ¬∃x P(x)

Answer 25

Copy

Question 26

(¬∃x P(x) → ∀x ¬P(x)) ∧ (∀x ¬P(x) → ¬∃x P(x))

Answer 26

Conjunction

Question 27

(¬∃x P(x) ↔ ∀x ¬P(x))…(a ↔ b) ≡ (a → b) ∧ (b → a)

∀P (¬∃x P(x) ↔ ∀x ¬P(x)).

Answer 27

Universal Generalization

Question 28

Assume ∀x P(x)

Answer 28

Assumption

Question 29

P(0)

Answer 29

Universal Instantiation

Question 30

∃x P(x)

Answer 30

Existential Generalization

Question 31

∴ ∀x P(x) → ∃x P(x)

Answer 31

Derivation