Uniqueness quantifier

denoted by ∃! or ∃1 The notation ∃!x P(x) [or ∃1xP(x)] states

“There exists a unique x such that P(x) is true.”

There is exactly one There is one and only one

Quantifiers with restricted domain

An abbreviated notation is used to restrict the domain of a quantifier

Example: What do the statements ∀x < 0 (x2 > 0), ∀y = 0 (y3 = 0),

and ∃z > 0 (z2 = 2) mean, where the domain in each case consists of the real numbers?

solution

The statement ∀x < 0 (x2 > 0) states that for every real number x with x < 0, x2 > 0.

That is, it states “The square of a negative real number is positive.” The statement ∀y = 0 (y3 = 0) states that for every real

number y with y = 0, we have y3 = 0. That is, it states “The cube of every nonzero real number is

nonzero.” Finally, the statement ∃z > 0 (z2 = 2) states that there exists a

real number z with z > 0 such that z2 = 2. That is, it states “There is a positive square root of 2.”

Precedence of qualifiers

The quantifiers ∀ and ∃ have higher precedence than all logical operators from propositional calculus.

Example, ∀xP(x) ∨ Q(x) is the disjunction of ∀xP(x) and Q(x).

In other words, it means (∀xP(x)) ∨ Q(x) rather than ∀x(P(x) ∨ Q(x)).

Binding variables

When a quantifier is used on the variable x, we say that this occurrence of the variable is bound.

An occurrence of a variable that is not bound by a quantifier or set equal to a particular value is said to be free.

All variables that occur in a propositional function must be bound or set equal to a particular value to turn it into a proposition

The part of logical expression to which a quantifier is applied is called the scope of this quantifier

example

∃x(x + y = 1), the variable x is bound by the existential quantification , but variable y is free because it is not bounded by any quantifier and has no value assigned to it. Hence x is bounded and y is free

∃x(P(x) ∧ Q(x)) ∨ ∀x R(x), here all variables are bounded. Scope of x in the first quantifier is different from the rest of the statement.

Logical equivalences using quantifiers Statements involving predicates and quantifiers

are logically equivalent if and 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.

Denoted by S ≡ T, where S and T are two statements involving predicates and quantifiers

example

Show that ∀x(P(x) ∧ Q(x)) and ∀x P(x) ∧ ∀x Q(x) are logically equivalent.

Negating quantifier expression-universal quantification￢∀ x P(x) ≡ ∃x ￢ P(x) Example “Every student in your class has taken a

course in calculus

solution

“It is not the case that every student in your class has taken a course in calculus.” or

“There is a student in your class who has not taken a course in calculus

Existential quantification

￢∃ x Q(x) ≡ ∀x ￢Q(x) “There is a student in this class who has

taken a course in calculus.”

solution

It is not the case that there is a student in this class who has taken a course in calculus

Every student in this class has not taken calculus

What are the negations of the statements “There is an honest politician” and “All Americans eat cheeseburgers”?

What are the negations of the statements ∀ x(x2 > x) and ∃x(x2 = 2)?

Show that ￢∀ x(P(x) → Q(x)) and ∃x(P(x)∧￢ Q(x)) are logically equivalent.

Translating English into logical expressions Express the statement “Every student in this

class has studied calculus” using predicates and quantifiers.

Express the statements “Some student in this class has visited Mexico” and “Every student in this class has visited either Canada or Mexico” using predicates and quantifiers.

Using quantifiers in system specifications Use predicates and quantifiers to express

the system specifications “Every mail message larger than one megabyte will be compressed” and “If a user is active, at least one network link will be available.”

Examples from Lewis Carroll

Consider these statements. The first two are called premises and the third is called the conclusion.The entire set is called an argument.

“All lions are fierce.” “Some lions do not drink coffee.” “Some fierce creatures do not drink coffee.”

Consider these statements, of which the first three are premises and the fourth is a valid conclusion.

“All hummingbirds are richly colored.” “No large birds live on honey.” “Birds that do not live on honey are dull in

color.” “Hummingbirds are small.”

Exercise problems

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))

homework

Translate these statements into English, where R(x) is “x is a rabbit” and H(x) is “x hops” and the domain consists of all animals.

a) ∀x(R(x) → H(x)) b) ∀x(R(x) ∧ H(x))c) ∃x(R(x) → H(x)) d) ∃x(R(x) ∧ H(x))

Exercise problem

Suppose that the domain of the propositional function P(x) consists of the integers −2, −1, 0, 1, and 2. Write out each of these propositions using disjunctions, conjunctions, and negations.

a) ∃xP(x) b) ∀xP(x) c) ∃x￢ P(x) d) ∀x ￢ P(x) e) ￢∃ xP(x) f ) ￢∀ xP(x)

Home work

Suppose that the domain of the propositional function P(x) consists of the integers 1, 2, 3, 4, and 5. Express these statements without using quantifiers, instead using only negations, disjunctions, and conjunctions.

a) ∃xP(x) b) ∀xP(x) c) ￢∃ xP(x) d) ￢∀ xP(x) e) ∀x((x = 3) → P(x)) ∨ ∃x￢ P(x)

Exercise problem

For each of these statements find a domain for which the statement is true and a domain for which the statement is false.

a) Everyone is studying discrete mathematics.b) Everyone is older than 21 years.c) Every two people have the same mother.d) No two different people have the same grandmother.

Home work

For each of these statements find a domain for which the statement is true and a domain for which the statement is false.

a) Everyone speaks Hindi.b) There is someone older than 21 years.c) Every two people have the same first name.d) Someone knows more than two other people.

Translate in two ways each of these statements into logical expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people.

a) Someone in your class can speak Hindi.b) Everyone in your class is friendly.c) There is a person in your class who was not born in California.d) A student in your class has been in a movie.e) No student in your class has taken a course in logic programming.

Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives.

a) No one is perfect.b) Not everyone is perfect.c) All your friends are perfect.d) At least one of your friends is perfecte) Everyone is your friend and is perfect.f ) Not everybody is your friend or someone is not perfect.

Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives.

a) Something is not in the correct place.b) All tools are in the correct place and are in excellent condition.c) Everything is in the correct place and in excellent condition.d) Nothing is in the correct place and is in excellent condition.e) One of your tools is not in the correct place, but it is in excellent condition.

Express each of these statements using logical operators, predicates, and quantifiers.

a) Some propositions are tautologies.b) The negation of a contradiction is a tautology.c) The disjunction of two contingencies can be a tautology.d) The conjunction of two tautologies is a tautology.