Predicate Logic

Question 1

What are the keywords for Universal Quantifiers ‘∀xP(x)’ ?

Answer 1

For all/any/every/each x

This is the conjunction of the propositions over all the elements

also known as ‘AND’

Question 2

What are the keywords for Existential Quantifiers ‘∃xP(x)’ ?

Answer 2

There exists /for some /for at least one x

This is the disjunction of the proposition over all the elements

also known as ‘OR’

Question 3

What are the keywords for Uniqueness Quantifiers ‘∃!xP(x)’ ?

Answer 3

There exist a unique x

∃! / ∃_1 : There exist exactly one

Question 4

[Nested Quantifiers]
∀x ∀y P(x, y)
meaning?

Answer 4

P(x, y) is true for every pair x, y.

Question 5

[Nested Quantifiers]
∃x ∃y P(x, y)
meaning?

Answer 5

There is a pair x, y for which P(x, y) is true.

Question 6

[Nested Quantifiers]
∃x ∀y P(x, y)
meaning?

Answer 6

There is an x for which P(x, y) is true for every y.

Question 7

[De Morgan’s Laws for Quantifiers]
What is the equivalent proposition to
∀x ( P(x) ∧ Q(x) )

Answer 7

∀x P(x) ∧ ∀x Q(x)

Question 8

[De Morgan’s Laws for Quantifiers]
What is the equivalent proposition to
∃x ( P(x) ∨ Q(x) )

Answer 8

∃x P(x) ∨ ∃x Q(x)

Question 9

State in order of precedence:
Logical Operators ( ∧, ∨, ¬ )
∀
∃

Answer 9

1. ∀
2. ∃
3. Logical Operators

Question 10

[Rules of inference]
What is Modus Ponens?

Answer 10

p → q
p
———-
∴ q

Question 11

[Rules of inference]
What is Modus Tollens?

Answer 11

¬q
p → q
————-
∴ ¬p

Question 12

[Rules of inference]
What is Conjunction?

Answer 12

p
q
———-
∴ p∧q

Question 13

[Rules of inference]
What is Simplification?

Answer 13

∴ p

Question 14

[Rules of inference]
What is Addition?

Answer 14

∴ p∨q

Question 15

[Rules of inference]
What is Hypothetical Syllogism?

Answer 15

p → q
q → r
————
∴ p → r

Question 16

[Rules of inference]
What is Disjunctive Syllogism?

Answer 16

p∨q
¬ p
————-
∴ q

Question 17

[Rules of inference]
What is Resolution?

Answer 17

p∨q
¬ p∨r
————-
∴ q∨r

Question 18

[Rules of inference with Quantifiers]
What is Universal Instantiation?

Answer 18

∴ P(c)

Question 19

[Rules of inference with Quantifiers]
What is Universal Generalisation?

Answer 19

∴ ∀xP(x)

Question 20

[Rules of inference with Quantifiers]
What is Existential Instantiation?

Answer 20

∴ P(c) for some element of D

Question 21

[Rules of inference with Quantifiers]
What is Existential Generalisation?

Answer 21

∴ ∃xP(x)

Question 22

[Rules of inference with Quantifiers]
What is Universal Modus Ponens?

Answer 22

∀xP(x) → Q(x)
P(a) for some element of D
————
∴ Q(a)

Question 23

[Rules of inference with Quantifiers]
What is Universal Modus Tollens?

Answer 23

∀xP(x) → Q(x)
¬ Q(x) for some element of D
————
∴ ¬ P(a)