1. Use quantifiers to express the following statements. Define your variables and propositional functions. Start by letting x be the set of students.

Every technology student needs a course in discrete mathematics.

Almost all students in this class owns a computer.

All the students in this class is an IST student or a sophomore.

There is not a single student in this class who has not used a computer.

The students in this class either like math or must take this math course.

2. Given the following propositions:
p: Jupiter's red spot is a giant pimple.
q: Saturn has rings around the collar.
r: Jupiter is a teen age planet.
Write the given argument in words and determine whether each argument is valid by indicating the applicable rule of inference.

p → r
r → q
p
\ q
3. Given the following:
p ↔ r
r
\ p
Use resolution to derive the conclusion. Replace ↔ with logically equivalent expressions that use or or ∨ and and or ∧.

4. Prove or disprove inductively that 2n + 1 ≤ 2n, n = 3, 4, . . ..

5. Prove or disprove inductively that any amount of money ≥ $0.05 that is a multiple of $0.05 can be formed using just nickels, dimes and quarters with the minimum number of nickels and dimes.

6. Show that x n, n ≥ 0 can be expressed recursively. What is the base case?