negations and variants of quantified statements

Negation of a universal statement

A universal statement is formally defined as: ∀ x ∈ D, p(x)

Where p(x) is a predicate and the domain of x is D.

Negation example: This can be read as "it is not the case that for all x in D, p(x) is true"

• There is at least 1 statement with the predicate negated

∃ x ∈ D such that ~p(x)

• Symbolically, this would be ~(∀ x ∈ D such that ~p(x)) == ∃ x ∈ D such that ~p(x)

Example: "All dogs are friendly"

• p(x) = "x is friendly"
• D = all dogs

Formally, ∀ x ∈ D, p(x)

The formal negation is: ∃ x ∈ D such that ~p(x)

• "There is a dog in D such that ~p(x)"
• "There is a dog that is not friendly"
• "Some dogs are not friendly"

Note: Negating an universal statement changes the qualifier and takes the negation of the predicate.

Negation of an existential statement

An existential statement is formally defined as: ∃ x ∈ D such that p(x)

Where p(x) is a predicate and the domain of x is D.

Negation example: This can be read as "for all x, p(x) is false" or "there is not an x that exists that is true"

∀ x ∈ D, ~p(x)

• Symbolically, this would be ~(∃ x ∈ D such that p(x)) == ∀ x ∈ D, ~p(x)

Note: Negating an existential statement changes the qualifier and takes the negation of the predicate.

Example: "there exists a question that is easy"

• s(x) = "x is easy"
• D = "all questions"

Formally, ∃ x ∈ D such that s(x)

The formal negation is: ∀ x ∈ D, ~s(x)

• "No questions are easy" or "all questions are not easy"

De Morgan's laws for negations of quantified statements

Universal statements:

• ~(∀ x ∈ D such that ~p(x)) == ∃ x ∈ D such that ~p(x)

Existential statements:

• ~(∃ x ∈ D such that p(x)) == ∀ x ∈ D, ~p(x)

We basically just swap the existential/universal quantifier and negate te predicate.

Negating a universal conditional statement

A universal conditional statement is of the form: ∀ x, if p(x) then q(x)

The negation of a universal conditional statement is formally defined as: ∃ x such that p(x) ^ ~q(x)

• Remember that the only way a universal conditional statement is False is to have

a True hypothesis with a False conclusion

• Symbolically, the logical equivalence is:

~(∀ x, if p(x) then q(x)) == ∃ x such that p(x) ^ ~q(x)

Example: "If a function is differentiable, then it is continuous"

• p(x) = "x is differentiable"
• q(x) = "x is continuous"
• F is the set of all functions

Formally, ∀ x, if p(x) then q(x)

The formal negation is: ∃ x such that p(x) ^ ~q(x)

• "There is at least one function that is differentiable but not continuous"
• "Some differentiable functions are not continuous"

Negation of an existential statement with conjunction

An existential with conjunction can be expressed as: ∃ x such that p(x) ^ q(x)

The negation of an existential statement is formally defined as: ∀ x, if p(x) then ~q(x)

• Symbolically, the logical equivalence is:

~(∃ x such that p(x) ^ q(x)) == (∀ x, if p(x) then ~q(x))

Example: "Some monkeys can speak French"

• m(x) = "x is a monkey"
• f(x) = "x speaks French"

Formally, ∃ x such that m(x) ^ f(x)

The informal negation is: (∀ x, if p(x) then ~q(x))

• "All monkeys can not speak French"

Variants of universal conditional statement

We know the most common variants of conditional statements, these definitions can be extended to universal conditional statements also.

Given the universal conditional statement of: ∀ x ∈ D , if p(x) then q(x)

• Contrapositive - ∀ x ∈ D , if ~q(x) then ~p(x)
• Converse - ∀ x ∈ D , if q(x) then p(x)
• Inverse - ∀ x ∈ D , if ~p(x) then ~q(x)

Example: Given - ∀ x ∈ R , if x > 3 then x^2 > 9

• p(x) = "x > 3"
• q(x) = "x^2 > 9"

Contrapositive:

• ∀ x ∈ R , if ~q(x) then ~p(x)
• "if a square of a real number is less than or equal to 9, then the number is

less than or equal to 3"

Converse:

• ∀ x ∈ D , if q(x) then p(x)
• "if the square of a real number is greater than 9, then the number is greater than 3"

Inverse:

• ∀ x ∈ D , if ~p(x) then ~q(x)
• "if a real number is less than or equal to 3, then the square of the numbe ris less

than or equal to 9"

Necessary condition

The definitions of necessary, sufficient, and only-if conditions can be extended to apply to universal conditional statements.

Given the sentence: ∀ x ∈ D, r(x) is a necessary condition for s(x)

Can be rewritten as: ∀ x ∈ D, if ~r(x) then ~s(x) == ∀ x ∈ D, if s(x) then r(x)

Example: "Passing a comprehensive exam is a necessary condition for obtaining a master's degree"

• m(x) = "x obtains a master's degree"
• r(x) = "x passes a comprehensive exam"
• D = set of all persons

Formally rewritten as: ∀ x ∈ D, if ~r(x) then ~m(x)

• Informally, "if a person does not pass a comprehensive exaxm then he/she does not obtain

a master's degree"

• == ∀ x ∈ D, if m(x) then r(x)
• Informally, "if a person obtains a master's degree then he/she passes a comprehensive exam"

Sufficient condition

Given the sentence: ∀ x ∈ D, r(x) is a sufficient condition for s(x)

Can be rewritten as: ∀ x ∈ D, if r(x) then s(x)

Example: "Squareness is a sufficient condition for rectangularity"

• p(x) = "x is a square"
• q(x) = "x is a rectangle"
• F = set of all plane figures

Formally rewritten as: ∀ x ∈ D, if p(x) then q(x)

• Informally, "if a plane figure is a square, then it is a rectangle"

Only if

Given the sentence: ∀ x ∈ D, r(x) only if s(x)

Can be rewritten as: ∀ x ∈ D, if r(x) then s(x) == x ∈ D, if ~s(x) then ~r(x)

Example: Given a statement - "A number is prime only if it is greater than 1"

• s(x) = "x is prime"
• r(x) = "x is greater than 1"
• D = set of all numbers

Formally rewritten as: ∀ x ∈ D, if r(x) then s(x)

• Informally, "if a number is prime, then it is greater than 1"
• == x ∈ D, if ~s(x) then ~r(x)
• Informally "if a number is less than or equal to 1, then it is not prime"