# Mathematics | Some theorems on Nested Quantifiers

Prerequisite – Predicates and Quantifiers – Set 1, Set 2

**Quantifiers** are expressions that indicate the scope of the term to which they are attached, here predicates. A *predicate* is a property the subject of the statement can have.

For example, in the statement *“the sum of x and y is greater than 5”*, the predicate ‘Q’ is- sum is greater than 5,

and the statement can be represented as Q(x, y) where x and y are variables.

The* scope *of a quantifier or a *quantification *is the range in the formula that the quantifier engages in.

**Types of quantification or scopes:**

**Universal(∀) –**The predicate is true for all values of x in the domain.**Existential(∃) –**The predicate is true for at least one x in the domain.

To know the scope of a quantifier in a formula, just make use of **Parse trees**. Two quantifiers are nested if one is within the scope of the other.

**Example-1:**∀x ∃y (x+y=5)

Here ‘∃’ (read as-there exists) and ‘∀’ (read as-for all) are quantifiers for variables x and y.

The statement can be represented as-

∀x Q(x)

Q(x) is ∃y P(x, y) Q(x)-the predicate is a function of only x because the quantifier applies only to variable x.

P(x, y) is (x + y = 5)**Example-2**

∀x ∀y ((x> 0)∧(y< 0) → (xy< 0))

(in English)

For every real number x and y, if x is positive and y is negative, implies xy is negative.

again,

∀x Q(x)

where Q(x) is ∀y P(x, y)

**Example to convert a statement into a nested quantifiers formula:**

“There is a pupil in this lecture who has taken at least one course in Discrete Maths.”

*A statement consists of quantifiers and predicates, split it into it's two constituents.*Here x and y are the pupil and the course and their respective quantifiers are attached in front of them.

Write it down as-

For some x pupil, there exist a course in Discrete Maths such that x has taken y.

∃x ∃y P (x, y), where P (x, y) is "x has taken y".

**Theorem-1:**The order of nested existential quantifiers can be changed without changing the meaning of the statement.**Theorem-2:**The order of nested universal quantifiers can be changed without changing the meaning of the statement.

**Example-3:**

Assume P(x, y) is xy=8,

∃x ∃y P(x, y) domain: integers

Translates to-

There is an integer x for which there is an integer y such that xy = 8,

which is same as-

There is a pair of integers x, y for which xy = 8.

Meaning ∃x ∃y P(x, y) is equivalent to ∃y ∃x P(x, y).Similarly,

Assume P(x, y) is (xy = yx).

∀x ∀y P(x, y) domain: real numbers

Translates to-

For all real numbers x, for all real numbers y, xy = yx or,

For every pair of real numbers x, y, xy = yx.

again ∀x ∀y P(x, y) is equivalent to ∀y ∀x P(x, y).However, when the nested quantifiers are not same, changing the order changes meaning of statement.

**Example-4:**Assume P(x, y, z) is (x + y = z).

∀x ∀y ∃z P(x, y, z) domain: real numbers

Translates to-

For all real numbers x and y there is a real number z such that x + y = z (True)

∀z ∃x ∃y P(x, y, z) domain: real numbers

There is a real number z such that for all real numbers x and y, x + y = z (False)

**Negation of nested quantifiers:**

**Theorem-3**

To negate a sequence of nested quantifiers, you change each quantifier in the sequence to the other type and then negate the predicate.

So the negation of ∀x ∃y : P(x, y) is ∃x ∀y : ~P(x, y)

**Example-5:**

“ ∃x at Cornell, x is at least 18 years old.”

To disagree with this, you’re negating the statement by flipping the ∃ to ∀ and then

negating the predicate:

“ ∀x at Cornell such that x is not at least 18 years old.”