8. 1.4.3. Existential Quantification

Try these first

1. 1.

   What is existential quantification?

   Hint

   Think of the statement involving 'there exists'.

2. 2.

   Write the notation for existential quantification.

   Hint

   What symbol represents 'there exists'?

3. 3.

   What does existential quantification assert?

   • All elements satisfy a property
   • At least one element satisfies a property
   • No elements satisfy a property

   Hint

   Think about the meaning of 'there exists'.

4. 4.

   Is the statement ∃x (x < 0) true if there are no negative numbers in the domain?

   • True
   • False

   Hint

   Consider what existence means in this scenario.

5. 5.

   Consider a domain of real numbers. Prove that if ∀x P(x) is false, then ∃x ¬P(x) is true.

   Hint

   Think about the definitions of universal and existential quantification.

6. 6.

   If you have a logical expression with both quantifiers, analyze how the scopes of the variables affect the meaning.

   Hint

   Analyze variable interactions under bounding conditions.