This document discusses logical expressions and quantifiers. It provides examples of using quantifiers like ∀ and ∃ to represent statements involving sets, predicates, and relationships between variables. Logical expressions are constructed using quantifiers, predicates with variables, and logical connectives to formally represent statements in English.

1. ‫آے‬ ‫وہ‬‫ہے‬ ‫قدرت‬ ‫کی‬ ‫خدا‬ ‫ہمارے‬ ‫میں‬ ‫گھر‬
‫ہ‬ ‫دیکھتے‬ ‫کو‬ ‫گھر‬ ‫اپنے‬ ‫کبھی‬ ‫انکو‬ ‫ہم‬ ‫کبھی‬‫یں‬
Slides-4

2. • It means at any time either I see my home or
my friend.
• It also concludes that I do not see somewhere
else and I do not see both at a time.
• Suppose I have many homes and many
friends but at least one friend is special one.
• Set of homes and friends are denoted by H
and F, respectively.
• Look(x) = Having a look at x
• ∃ℎ ∈ 𝐻∃𝑓 ∈ 𝐹: 𝐿𝑜𝑜𝑘 ℎ ⊕ 𝐿𝑜𝑜𝑘(𝑓)

3. Exercise
• How to modify the formula in previous slide if
I say we are talking about the home in which
friend is there?
• Write a logical expression for “All the time if
door is closed then I am not inside my cabin”.
Where the propositions p and q are “Door is
closed” and “I am inside my cabin”.
• “All the time if door is closed then I am not
inside my cabin otherwise I am inside”.

4. Excecise
Translate these statements into English, where
C(x) is “x
is a comedian” and F(x) is “x is funny” and the
domain
consists of all people.
a) ∀x(C(x) → F(x)) b) ∀x(C(x) ∧ F(x))
c) ∃x(C(x) → F(x)) d) ∃x(C(x) ∧ F(x))

5. Exercise
• P(x): “x is a professor”
• I(x): “x is ignorant”
• V(x): “x is vain”
• Express the following statements using
quantifiers, logical connectives and P(x), I(x), and
V(x), where the universe is the set of all people.
• a) No professors are ignorant.
• b) All ignorant people are vain.
• c) No professors are vain

6. Nested Quantifiers
 xy P(x, y)
 “For all x, there exists a y such that P(x,y)”.
 Example: xy (x+y == 0) where x and y are integers
 xy P(x,y)
 There exists an x such that for all y P(x,y) is true”
 xy (x*y == 0)

7. Meanings of multiple quantifiers
 xy P(x,y)
 xy P(x,y)
 xy P(x,y)
 xy P(x,y)
P(x,y) true for all x, y pairs.
For every value of x we can find a (possibly different)
y so that P(x,y) is true.
P(x,y) true for at least one x, y pair.
There is at least one x for which P(x,y)
is always true.
quantification order is not
commutative.
Suppose P(x,y) = “x’s favorite class is y.”

8. Example
• Let S be the set of students, i.e., {s1, s2, s3, …}
and P be the set of sports {hockey, cricket,
badminton, tennis, chess}
• Let Q (x,y):= “x plays y”
be a predicate which is true if x plays y
otherwise false.
• Then
xy Q(x,y) means every student plays each
game

9. • xy Q(x,y) There is at least one student from
S who plays at least one game from the set P.
• xy Q(x,y) All students play at least one
game.
• xy Q(x,y) There is at least one student who
plays all games.

10. Exercise
• Let M(x), F(x), E(x) and S(x,y) be the
statements for “x is a male”, “x is a female”, “x
is employee of COMSATS” and “x and y are
spouse”, respectively.
• Write a logical formula for “There is at least
one couple in employees of COMSATS and one
of them is male and other is female”.

11. Bound and free variables
A variable is bound if it is known or
quantified. Otherwise, it is free.
Examples:
P(x) x is free
P(5) x is bound to 5
x P(x) x is bound by quantifier
Reminder: in a
proposition, all
variables must
be bound.