Predicates and Quantifiers

Question 1

Define variables

Answer 1

To denote subjects

Question 2

Define predicates

Answer 2

to denote a property of a subject

Question 3

Define quantifiers

Answer 3

To quantify over subjects

Question 4

What is a universal Quantifier (∀x)

Answer 4

asserts P(x) is true for every value of x in the domain

Question 5

What is an Existential Quantifier (∃x)

Answer 5

assers P(x) is true for some values of x in the domain

Question 6

Example (Quantifiers): If P(x) denotes “x is even” and U then integer then:

Answer 6

∀x. P(x) is false, but
∃x. P(x) is true.

Question 7

What precedence do the ∀x ∃x have over other logical operators

Answer 7

They have a higher precedence than all the logical operators

Question 8

What do truth values depend on

Answer 8

The predicate (P(x)) and the domain (U)

Question 9

If U is the positive integers and P(x) is the statement “x < 2” then what are the truth values for ∀x and ∃x

Answer 9

For ∃x P(x) is true but for ∀x P(x) is false

Question 10

What is very important in nested quantifiers

Answer 10

the order (the first two doesn’t matter but the second 2 does (2,3)

Question 11

Define argument in propositional logic

Answer 11

a sequence of propositions

Question 12

When is the argument valid

Answer 12

when the premises imply the conclusion

Question 13

What is the last statement called in proportional logic

Answer 13

the conclusion

Question 14

What does this statement in english ∀x∀y P(x, y) ≡ ∀x∀y (x + y = 0)

Answer 14

For all real numbers x and y, x + y = 0

Question 15

ExEy what does this mean in English

Answer 15

There exists x, there exists y

Question 16

p –> q, what is a trivial proof in this case

Answer 16

if q is true

Question 17

p –> q, what is a vacuous proof in this case

Answer 17

if p is false

Question 18

p –> q, what is a direct proof in this case

Answer 18

Assume p then show q

Question 19

p –> q, what is proof by contraposition in this case

Answer 19

¬q and show ¬p

Question 20

Example of direct proof: If n is an odd integer, then n^2 is odd, n = 2k +1

Answer 20

Proof. Assume that n is odd.
* Then n = 2k + 1 for an integer k.
* Thus,
n
= (2k + 1)^2 = 4k^2 + 4k + 1
= 2(2k^2 + 2k) +1
= 2r + 1

therefore always odd