Mathematics | Some Theorems on Nested Quantifiers

Quantifiers are expressions that indicate the scope of the term to which they are attached, they are 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

1. Universal (∀) - The predicate is true for all values of x in the domain.
2. 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)

Solved Examples - Some Theorems on Nested Quantifiers

Example 1

Statement: (\forall x \exists y , (x + y = 10))

Solution:

For every ( x ), there exists a ( y ) such that ( x + y = 10 ). Choosing ( y = 10 - x ) will satisfy the equation for any ( x ).

Example 2

Statement: (\exists y \forall x , (x + y > x)) Solution: There exists a ( y ) such that for every ( x ), ( x + y > x ). Choosing any ( y > 0 ) will satisfy this condition because adding a positive number ( y ) to any ( x ) will always result in a number greater than ( x ).

Example 3

Statement: (\forall x \exists y , (x \cdot y = 1))

Solution: For every ( x ) (where ( x \neq 0 )), there exists a ( y ) such that ( x \cdot y = 1 ). Choosing ( y = \frac{1}{x} ) will satisfy the equation.

Practice Problems - Some Theorems on Nested Quantifiers

1. (\forall x \exists y , (x2 + y2 = 1))

2. (\exists y \forall x , (x + y \geq 0))

3. (\forall x \exists y , (xy = x + y))

4. (\exists y \forall x , (x - y \leq x))

5. (\forall x \exists y , (x3 + y = 0))

6. (\exists y \forall x , (x2 + y \geq 1))

7. (\forall x \exists y , (x + y > 1))

8. (\exists y \forall x , (x - y < 0))8

9. (\forall x \exists y , (x^2 + xy = y))

10. (\exists y \forall x , (x + y \leq 1))

Related Articles:

• Introduction to Mathematical Logic
• Demorgans law
• Predicates and Quantifiers