Quiz 21

Question 1

How do you use (∃x∈A) P(x) in a proof?

Answer 1

Use x∈A where x is generic. Use P(x).

Question 2

What is the following inference rule called?

P(x) where x can be anything you choose

∴ ∃x P(x)

Answer 2

Existential Generalization

Question 3

Select a reasonable strategy for proving ∀x P(x)

Answer 3

Prove P(x) where x is generic.

Question 4

Which of the following is called Universal Instantiation?

Answer 4

∀x P(x)

∴ P(x) where x can be anything you choose

Question 5

Select all reasonable strategies for proving q → p

Answer 5

Assume q. Prove p.
Assume ¬p. Prove ¬q
Assume q. Assume ¬p. Prove 0.

Question 6

What is the following inference rule called?

x∈A → P(x) where x is generic

∴ (∀x∈A) P(x)

Answer 6

Universal Generalization

Question 7

How might you use ∃x P(x) in a proof?

Answer 7

Existential Instantiation

Question 8

Which rule of inference justifies the final step in a typical proof of (∀x∈A) P(x)?

Answer 8

Universal Generalization

Question 9

How might you use p → q in a proof of p?

Answer 9

You can’t.

Question 10

Select a reasonable strategy for proving (p → q) → r

Answer 10

Assume p → q. Prove r.

Question 11

Select all reasonable strategies for proving p.

Answer 11

Use the definition of p.
Assume ¬p. Prove 0.
Assume ¬p. Prove p.

Question 12

How do you use ∀x P(x) in a proof?

Answer 12

Use P(x) where x is whatever you want it to be.

Question 13

Select all reasonable strategies for proving (p ∨ q) → r

Answer 13

Prove p → r.  Prove q → r.
Part 1: Assume p.  Prove r.
Part 2: Assume q.  Prove r.
Part 1: Assume q.  Prove r.
Part 2: Assume p.  Prove r.

Question 14

Select a reasonable strategy for proving (∃x∈A) P(x)

Answer 14

Prove x∈A where x is anything you want it to be. Prove P(x).

Question 15

Which of the following is called Existential Instantiation?

Answer 15

∃x P(x)

∴ P(x) where x is generic

Question 16

Select all reasonable strategies for proving p ↔ q.

Answer 16

Prove p → q.  Prove q → p.
Prove p → q.  Prove ¬p → ¬q.
Part 1: Assume p.  Prove q.
Part 2: Assume q.  Prove p.
Part 1: Assume q.  Prove p.
Part 2: Assume p.  Prove q.

Question 17

Select all reasonable strategies for proving (∀x∈A) P(x)

Answer 17

Prove x∈A → P(x), where x is generic.

Assume x∈A where x is generic. Prove P(x).

Question 18

How might you use p → q in a proof of ¬p?

Answer 18

Modus tollens.

Question 19

Which rule of inference justifies the final step in a typical proof of ∃x P(x)?

Answer 19

Existential Generalization

Question 20

How might you use (∃x∈A) P(x) in a proof?

Answer 20

Existential Instantiation

Question 21

What is the following inference rule called?

(∃x∈A) P(x)

∴ x∈A ∧ P(x) where x is generic

Answer 21

Existential Instantiation

Question 22

Select a reasonable strategy for proving ∃x P(x)

Answer 22

Prove P(x) where x is whatever you want it to be.

Question 23

What is the following inference rule called?

(∀x∈A) P(x)

x∈A where x can be anything you choose

∴ P(x)

Answer 23

Universal Instantiation

Universal Modus Ponens

Question 24

How might you use ∀x P(x) in a proof?

Answer 24

Universal Instantiation

Question 25

Select a reasonable strategy for proving (p ∧ q) → r

Answer 25

Assume p. Assume q. Prove r.

Question 26

Select a reasonable strategy for proving p ∧ q

Answer 26

Prove p. Prove q.

Question 27

How might you use p → q in a proof of q?

Answer 27

Modus ponens.

Question 28

What is the following inference rule called? P(x) where x is generic ∴ ∀x P(x)

Answer 28

Universal Generalization

Question 29

Which of the following is called Existential Generalization?

Answer 29

P(x) where x can be anything you choose ∴ ∃x P(x)

Question 30

How do you use p ∧ q in a proof?

Answer 30

Use p. Use q.

Question 31

What is the following inference rule called? P(x) where x can be anything you choose x∈A ∴ (∃x∈A) P(x)

Answer 31

Existential Generalization

Question 32

Select all reasonable strategies for proving ¬p.

Answer 32

Use the definition of p. Assume p. Prove 0. Assume p. Prove ¬p.

Question 33

Which of the following is called Universal Generalization?

Answer 33

P(x) where x is generic ∴ ∀x P(x)

Question 34

Which of the following are valid inference rules?

Answer 34

First 4 ∀x P(x) ∴ P(x) where x can be anything you choose P(x) where x is generic ∴ ∀x P(x) ∃x P(x) ∴ P(x) where x is generic P(x) where x can be anything you choose ∴ ∃x P(x)

Question 35

How do you use ∃x P(x) in a proof?

Answer 35

Use P(x) where x is generic

Question 36

Which rule of inference justifies the final step in a typical proof of ∀x P(x)?

Answer 36

Universal Generalization

Question 37

What is the following inference rule called? ∃x P(x) ∴ P(x) where x is generic

Answer 37

Existential Instantiation

Question 38

What is the following inference rule called? ∀x P(x) ∴ P(x) where x can be anything you choose

Answer 38

Universal Instantiation

Question 39

Which rule of inference justifies the final step in a typical proof of (∃x∈A) P(x)?

Answer 39

Existential Generalization

Question 40

How do you use (∀x∈A) P(x) in a proof?

Answer 40

Prove x∈A where x is anything you want it to be. Use P(x).

Question 41

How might you use p → q in a proof of ¬q?

Answer 41

You can’t.