Lecture 3

Question 1

what is a predicate ?

Answer 1

its a propositional function) ;
- a descriptive sentence that could contain some variables.
- neither true nor false when the values of the variables are not specified.

Question 2

The statement P(x) is also said to be the …… of the propositional function P at x, when the value of x is declared , the propositional function p(x) becomes a ……. and have ……

Answer 2

value
proposition , (and have truth values)

Question 3

• The statement “x is greater than 3” has two parts.
  The first part:……………, is………….. of the statement.
  The second part: …………., “is greater than 3”—refers to a………… that the subject of the statement can have.
• We can denote the statement “x is greater than 3” by ……..,
  where …………… “is greater than 3” and x is the variable.

Answer 3

the variable x is the subject of the statement.
the predicate is property that the subject of the statement can have

• P(x) , P denotes the predicate

Question 4

what is quantification ?

Answer 4

Quantification is another method for obtaining a proposition from the propositional function .
- it expresses the extent to which a predicate is true over a range of elements.
- the words all, some, many, none, and few are
used in quantifications.

Question 5

what is a universal quantification ?

Answer 5

• The universal quantification of P(x) for a particular domain is the proposition that asserts that P(x) is TRUE FOR ALL VALUES of X IN THIS DOMAIN.
• “P(x) for all values of x in the domain.”
• The domain must always be specified when a universal quantifier is used; The meaning of the universal quantification of P(x) changes when we change the domain.

Question 6

We read ∀xP(x) as “……………..OR ………………
An element for which P(x) is false is called a …………… to ∀xP(x).

Answer 6

• “for all xP(x)” or “for every xP(x)”
• counterexample to ∀xP(x).

Question 7

when is ∀xP(x) TRUE ?

Answer 7

when all x’s in the domain satisfies P(x)

(counterexample) it’s false if there exists an x in the domain that doesn’t satisfy P(x).

Question 8

when is ∃xP(x) is TRUE ?

Answer 8

when there exists an x that satisfies P(x).

(counterexample) it’s false if all x’s in the domain is NOT suitable for P(x) .

Question 9

what is existential quantification ?

Answer 9

∃xP(x)
- is the proposition “There exists an element x in the domain
such that P(x).”

Question 10

what is uniqueness quantification ?

Answer 10

• ∃! xP(x) or ∃1 xP(x)
• “There exists a unique x such that P(x) is true.”
• “there is exactly one x that satisfies p(x)”
• “there is one and only one x that satisfies p(x).”

Question 11

when can we express quantified statements using propositional logic ?

Answer 11

• When the domain of a quantifier is finite, that is, when all
  its elements can be listed.
• ie :- we can express the universal quantification ∀xP(x) as the conjunction P(x1) ∧ P(x2) ∧⋯∧ P(xn),
  :- we can express the existential quantification ∃xP(x) as the disjunction P(x1) ∨ P(x2) ∨⋯∨ P(xn),

Question 12

when can we say that two statements involving predicates and quantifiers are “ logically equivalent “ ? S1 ≡ S2

Answer 12

IF & ONLY IF they have the same truth value no matter which - predicates are substituted into these statements and
- which domain of discourse is used for the variables in these propositional functions.

Question 13

we can distribute a universal quantifier over a …………
we can also distribute an ………… over a disjunction.

Answer 13

conjunction.
existential quantifier

Question 14

how do you negate Quantified Expressions with example ?

Answer 14

using De Morgan’s laws for quantifiers. :-
1) ∀xP(x) negation will be ∃x ¬P(x).
ex:- if ∀xP(x) implies that “All students in your class has taken calculus.”
then its negation will happen if NOT ALL students have taken calculus ie:- There exists a student in this class who has NOT taken calculus

2) ∃xP(x) negation will be ∀x¬P(x)
ex:- if ∃xP(x) implies that “ there exists a student who failed this class “
then its negation will happen if this particular student did NOT fail this class ie:- ALL students in this class did NOT fail ∀x¬P(x)

Question 15

what’s the difference between a premises, conclusion & an argument?

Answer 15

• An argument is a series of statements consisting of premises and a conclusion
• argument is a set of reasons offered in support of a claim.
• A statement is a premise (sentence which has a truth value) in an argument.

1. If it snows, then it’s cold (premise)
2. It snows (premise)
3. Therefore, it is cold (conclusion)

Question 16

“Some lions do not drink coffee.”
P(x) = “x is a lion”
R(x) = “x drinks coffee”

what’s the correct implementation for this ? why cant we use implication?

Answer 16

∃x(P(x) ∧ ¬R(x)).
using implication ∃x(P(x) → ¬R(x)).
would be incorrect because it implies :-
1) that the statement is still valid if x is not a lion, (truth table for implication : false implies everything which is not the case)
2) there is at least one creature that is not a lion,