Nested quantifiers are quantifiers used together in a statement, where one quantifier comes inside the scope of another quantifier. They help describe relationships between different variables.
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.
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, then xy is negative. again, ∀x Q(x) where Q(x) is ∀y P(x, y).
Converting Statements into Nested Quantifier Formulas
A statement in predicate logic is written using quantifiers and predicates.
- Quantifiers: ∀ (for all), ∃ (there exists).
- Predicate: a property or relation involving variables.
To convert a statement into a nested quantifier formula:
- Identify the variables
- Identify the predicate
- Attach suitable quantifiers
Example: “There is a student in this lecture who has taken at least one course in Discrete Mathematics.”
Let:
- x = Student
- y = Discrete Mathematics course
- P(x, y) = “x has taken y”
The statement becomes: “There exists a student x and there exists a course y such that x has taken y.”
Symbolically: ∃x ∃y P(x, y)
Theorems on Nested Quantifiers
Theorem 1: The order of nested existential quantifiers can be changed without changing the meaning of the statement.
∃x ∃y P(x, y) ≡ ∃y ∃x P(x, y)
Example: Statement: P(x, y): xy = 8
Domain: Integers
The statement means: “There exist integers x and y such that xy = 8.”
Changing the order of quantifiers gives the same meaning.
Hence, ∃x ∃y P(x, y) ≡ ∃y ∃x P(x, y).
Theorem 2: The order of nested universal quantifiers can also be changed without changing the meaning of the statement.
∀x ∀y P(x, y) ≡ ∀y ∀x P(x, y)
Example: Statement: ∀x ∀y (xy = yx)
Domain: Real Numbers
The statement means: “For all real numbers x and y, xy = yx.”
Changing the order of quantifiers gives the same meaning.
Hence, ∀x ∀y P(x, y) ≡ ∀y ∀x P(x, y).
Theorem 3: To negate nested quantifiers, replace each quantifier with its opposite type (∀ becomes ∃ and ∃ becomes ∀) and negate the predicate. Thus, the negation of ∀x ∃y P(x, y) is ∃x ∀y ~P(x, y).
Example: Statement: ∀x ∃y (x + y = 0)
Domain: Integers
The statement means: “For every integer x, there exists an integer y such that x + y = 0.”
Its negation is: ∃x ∀y (x + y ≠ 0)
which means: “There exists an integer x such that for every integer y, x + y ≠ 0.”
Solved Examples
Example 1: Statement: (∀x ∃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: (∃y ∀x (x + y > x))
Solution:
“There exists a value of y such that for every value of x, x + y > x.”
Choosing any positive value of y (y > 0) satisfies the condition because adding a positive number to x always gives a value greater than x.
Example 3: Statement: (∀x ∃y (x · y = 1))
Solution:
“For every value of x (where x ≠ 0), there exists a value of y such that x · y = 1.”
Choosing, y = 1/x
satisfies the equation because: x · (1/x) = 1
Practice Problems
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