Logical Quantifier

Logical quantifiers apply a predicate to elements in a domain or set. They can be thought of as a more generalised version of Summation and product notation ($\Sigma ,\Pi$)

Universal Quantifier §

See Universal Quantifier

Existential Quantifier §

See Existential Quantifier

Quantifier Scope §

See Quantifier Scope

Properties §

• Both quantifiers are commutative with themselves only:
  • $\forall x\forall y\equiv \forall y\forall x$
  • $\exists x\exists y\equiv \exists y\exists x$
  • $\forall x\exists y\not\equiv \exists y\forall x$

Quantifier order

The order of quantifiers can greatly change a statement! Look at the two examples:

• $\forall x\in \mathbb{R},(\exists y\in \mathbb{R},x+y=0)$
• $\exists y\in \mathbb{R},(\forall x\in \mathbb{R},x+y=0)$

The first statement states that for every real number, there exists another number such that the sum of the two is zero. This is the existence of the additive inverse and is true as the real numbers are a field. However, the second statement says that there exists a real number that can be added to every other real number (including itself) and return 0. Such a number does not exist!

Negating Quantifiers §

$$\neg (\forall x\in A,P(x))\equiv \exists x\in A,\neg P(x)$$

$$\neg (\exists x\in A,P(x))\equiv \forall x\in A,\neg P(x)$$

This property is used extensively to disprove a statement. For example, to disprove the statement ‘All odd integers are prime’ we need to show that the negation is true, i.e. we need to show that there exists an odd integer that is not prime. Similarly, for the statement ‘There exists a real number with square < 0’, we need to show the negation is true, that is, we need to show that all real numbers have square >= 0.