Nested Quantifiers Practice Questions

Nested quantifiers are two or more quantifiers used together in a logical statement, where one quantifier is placed inside the scope of another. They are used to describe relationships between multiple variables, and the order of the quantifiers affects the meaning of the statement.

Example:

Consider the statement:

∀x ∈ {1, 2}, ∃y ∈ {2, 4} such that y > x

This means "For every element x in the set {1, 2}, there exists an element y in the set {2, 4} that is greater than x."

• For x = 1, choose y = 2.
• For x = 2, choose y = 4.

Since a suitable y exists for every x, the statement is true.

Question 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 ).

Question 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.

Question 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

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

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

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