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- Computer Science Department - Rutgers University Spring 2020
- CS 205: Homework Set 2 16:198:205
- Complete each of the following problems to the best of your ability. Remember, you can’t be graded on what you
- don’t write down so (unless you are just making stuff up) something is better than nothing. Discussing problems
- between each other is fine, but your final writeup and work must be your own.
- • Inference
- 1) The Resolution Inference Rule states that if (P ∨ Q) and (¬P ∨ R), we may conclude (Q ∨ R). Show
- (P ∨ Q) ∧ (¬P ∨ R) → (Q ∨ R) (1)
- is a tautology, i.e., always true. Try to do it without a truth table - could you use any other rules of
- 2) Modus Ponens is the rule that if P and P → Q, we may conclude Q. Show that this is really a simple
- version of the resolution inference rule. Hint: Consider a very specific R.
- 3) It can be shown that the resolution inference rule is ‘complete’ in the sense that it is all that is needed
- in order to conclude or infer everything that can be concluded or inferred. Why might other rules of
- inference still be interesting or useful?
- 4) (1.6.20 in the book) Determine whether these are valid arguments. Abstract the statements as much as
- possible to be able to analyze the logical structure / errors.
- ∗ If x is a positive real number, then x
- is a positive real number. Therefore, if a
- is positive, where a
- is a real number, then a is a positive real number.
- ∗ If x
- 2 6= 0 where x is a real number, then x 6= 0. Let a be a real number with a
- 2 6= 0; then a 6= 0.
- 5) (1.6.26) Justify the rule of universal transitivity, which states that if ∀x : P(x) → Q(x) and ∀x :
- Q(x) → R(x) are true, then ∀x : P(x) → R(x).
- 6) Someone shows you four cards, each with a letter on one side and a number on the other, and says the
- following: “Any card with a vowel on one side has an even number on the other side.” The four cards you
- are shown reveal the following: A, 7, B, 4. If you wanted to verify their claim, which cards would you
- flip over, and why?
- • Proofs
- 1) (1.7.16) Prove that if m and n are integers and nm is even, then m is even or n is even.
- ∗ What is the best approach here, direct proof, proof by contraposition, or proof by contradiction -
- ∗ Complete the proof.
- 2) Prove that for any integer n, n is divisible by 3 iff n
- is divisible by 3. Does your proof work for divisibility
- by 4 - why or why not? Identify the kind of proof steps you used, and why.
- 3) Prove that √
- 3 is irrational. What is the best proof approach to take, and why?
- 4) Prove that if you take 101 pigeons, and try to force them into 100 pigeonholes, there is some hole that
- has two pigeons. What is the best proof approach to take, and why?
- 5) Generalize the result of the previous problem: that if you take N pigeons and try to force them into
- M < N pigeonholes, some hole must have at least N/M many pigeons.
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