13.4.2. Part b

Try these first

1. 1.

   Define a predicate to represent 'x is a student'.

   Hint

   Think of how to state who belongs to which category.

2. 2.

   What does the statement ∀x(M(x) → W(x)) imply?

   Hint

   Focus on breaking down the elements of the statement.

3. 3.

   What does ∃x(S(x) ∧ W(x)) mean?

   • No students have left
   • At least one student is both a math major and left
   • All math majors have left

   Hint

   Focus on the existential meaning of the statement.

4. 4.

   Is it true that if P(x) is true, then P(x+1) must be true?

   • True
   • False

   Hint

   Think of exceptions to the statement.

5. 5.

   Construct a logical argument involving predicates for a scenario where all seniors earn higher than 3.0, but not all students do. Determine its validity.

   Hint

   Break it down into parts, analyze the components.

6. 6.

   Create an example showing P(0) is true and ∀n(P(n) → P(n+1)) is invalid.

   Hint

   Identify properties that diverge.