Quantifiers and Negation

Quantifiers and negation are fundamental concepts in logic, mathematics, and computer science, particularly in predicate logic.

• Quantifiers specify the quantity of instances for which a predicate is true.
• Negation is the logical operation that reverses the truth value of a statement. If a statement P is true, then its negation ¬P is false, and vice versa.

Let's discuss these in detail.

Quantifier Definition

Quantifiers are symbols or words used in logical statements to indicate the quantity of elements being referred to. They are essential in forming statements in mathematical logic and set theory.

There are two types of quantifiers:

• Universal Quantifier
• Existential Quantifier

Universal Quantifier ( ∀ )

The symbol ∀ means "for all" or "for every." It asserts that a property holds true for every element in a specified set.

For example, the statement \forall x \in \mathbb{R}, x^2 \geq 0 means "for all real numbers x, x² is non-negative."

Existential Quantifier ( ∃ )

The symbol ∃ means "there exists" or "there is at least one." It indicates that there is at least one element in a specified set for which the property holds.

For instance, the statement \exists x \in \mathbb{Z} \text{ such that }  x²=4 means "there exists an integer x such that x² equals 4."

Examples of Quantifiers

• Universal: \forall n \in \mathbb{N}, n + 1 > n (Every natural number is less than its successor).
• Existential: \exists n \in \mathbb{N} \text{ such that } n \text{ is even} (There exists a natural number that is even).

Negation Definition

Negation is a logical operation that reverses the truth value of a statement. If a statement is true, its negation is false, and vice versa. Understanding negation is vital for logical reasoning and proof construction.

Examples of Negation

• Original Statement: \forall x \in \mathbb{R}, x^2 > 0 (For all real numbers x, x² is positive).
  Negated Statement: \exists x \in \mathbb{R} \text{ such that } x^2 \leq 0 (There exists a real number x such that x² is non-positive).

• Original Statement: \exists y \in \mathbb{Z} \text{ such that } y + 1 = 0 (There exists an integer y such that y + 1 = 0).
  Negated Statement: \forall y \in \mathbb{Z}, y + 1 \neq 0 (For all integers y, y + 1 is not zero).

How to Negate Statements

To negate a statement, we typically use the following rules:

• Negate a universal statement by replacing it with an existential statement: \neg ( \forall x, P(x) ) \equiv \exists x \text{ such that } \neg P(x)
• Negate an existential statement by replacing it with a universal statement: \neg ( \exists x, P(x) ) \equiv \forall x, \neg P(x)

Interpreting Statements with Quantifiers and Negation

Using quantifiers with negation allows us to express more complex logical statements. For example, the negation of "All birds can fly" translates to "There exists at least one bird that cannot fly."

• Original: \forall x \in \mathbb{R}, x^2 \geq 0 (All real numbers are non-negative).
  Negation: \exists x \in \mathbb{R} \text{ such that } x^2 < 0 (There exists a real number such that x2x^2x2 is negative).

• Original: \exists n \in \mathbb{N} \text{ such that } n \text{ is prime} (There exists a natural number that is prime).
  Negation: \forall n \in \mathbb{N}, n \text{ is not prime} (Every natural number is not prime).

Solved Examples on Quantifiers and Negation

Example 1: Negating a Universal Statement

Original Statement: \forall n \in \mathbb{Z}, n + 1 > n

Solution:

Negated Statement: \exists n \in \mathbb{Z} \text{ such that } n + 1 \leq n

This means there exists an integer n such that n + 1 is not greater than n.

Example 2: Negating an Existential Statement

Original Statement: \exists x \in \mathbb{R} \text{ such that } x^2 = 4

Solution:

Negated Statement: \forall x \in \mathbb{R}, x^2 \neq 4

This means that for every real number x, x² is not equal to 4.

Example 3: Negating the Universal Quantifier

Original Statement: \forall x \in \mathbb{N}, x + 1 \geq 2

Solution:

Negated Statement: \exists x \in \mathbb{N} \text{ such that } x + 1 < 2
This means there exists at least one natural number x such that x + 1 is less than 2.

Example 4: Negating the Existential Quantifier

Original Statement: \exists y \in \mathbb{Z} \text{ such that } y^2 = -1

Solution:

Negated Statement: \forall y \in \mathbb{Z}, y^2 \neq -1
This means that for every integer y, y² is not equal to -1.

Example 5: Negating the Combined Quantifiers

Original Statement: \forall x \in \mathbb{R}, \exists y \in \mathbb{R} \text{ such that } y = x^2

Solution:

Negated Statement: \exists x \in \mathbb{R} \text{ such that } \forall y \in \mathbb{R}, y \neq x^2
This means there exists a real number x such that for every real number y, y is not equal to x².

Practice Questions on Quantifiers and Negation

You can download this free pdf with answer key from the following:

Worksheet on Quantifiers and Negation - Free PDF Download