8. 1.4.2. Universal Quantification

Try these first

1. 1.

   Explain what universal quantification means.

   Hint

   Think about the notation ∀x.

2. 2.

   Provide a simple example of a universally quantified statement.

   Hint

   Consider what numbers are included in the natural numbers.

3. 3.

   What does the notation ∀x, P(x) represent?

   • For all x
   • P(x) is true
   • There exists an x such that P(x) is true
   • P(x) holds for some x

   Hint

   Remember the symbol used!

4. 4.

   True or False: Universal quantification is the same as existential quantification.

   • True
   • False

   Hint

   Think about what each quantifier states.

5. 5.

   Consider the predicate P(x): 'x is a square number'. What can you deduce about ∀x, P(x) in the domain of all integers?

   Hint

   Consider known square numbers.

6. 6.

   Prove or disprove: ∀x, (x + 2 > x) is true for all integers.

   Hint

   Think about the properties of integers.