CmSc180 Discrete Mathematics
Homework 04 due 02/10
- Give direct proof for the following statements
1.a. The sum of two odd numbers is even
1.b. The sum of an even and an odd number is odd
When constructing the proof, follow the example below:
Prove that the sum of two even numbers is even.
Proof:
Let A and B be two arbitrary chosen even numbers
(1) x, even(x) multiple of 2(x), i.e. p, integer(p) & x = 2p
(2) even(A)given in the problem
(3) even(B) given in the problem
(4) p, integer(p) such that A = 2p by (1), (2) and MP
(5) q, integer(q) such that B = 2q by (1), (3) and MP
(6) S = A + B = 2p + 2q = 2(p+q) by (4), (5), and basic algebra
(7) x, multiple of 2(x) even(x) by definition of even numbers
(8) multiple_of_2(S) by (6)
(9) even(S)by (7), (8) and MP
- Using the predicates student(x), study(x), play_soccer(x), healthy (x) and appropriate quantifiers (,), represent in predicate logic the following sentences, write the negation of the predicate expression and translate back to English
2.1.All students study.
2.2.Some students play soccer and study
2.3.Some soccer players are not students
2.4.Students that play soccer are healthy.
2.5.Some healthy students play soccer
2.6.Some healthy soccer players are students
2.7.All healthy soccer players are students
2.8.All soccer players are healthy students
2.9.All students are healthy soccer players
- Use propositional logic to prove that the following argument is a valid argument.
If my client is guilty, then the knife was in the drawer. Either the knife was not in the drawer, or Jason Pritchard saw the knife. If the knife was not there on October 10, it follows that Jason Pritchard did not see the knife. Furthermore, if the knife was there on October 10, then the knife was in the drawer and also the hammer was in the barn. But we all know that the hammer was not in the barn. Therefore, ladies and gentlemen of the jury, my client is innocent.
Note to problem 3: Here we do not question the truth of the statements in the argument. We assume that they are true. The question we have to answer is: is the argument a valid argument. When solving this problem, follow the “Sherlock Holmes” example solved in class.
- Represent the following arguments in predicate logic and determine whether they are valid or invalid. If valid determine the type of argument. If invalid determine the type of error if the type is known.
All nerds are good at math.
Buffy is good at math.
Buffy is a nerd.
It is difficult to study whenever I am tired.
I found it easy to study today.
I am not tired today.
Animals at the bottom of the food chain are very nervous.
People are not at the bottom of the food chain.
People are not nervous
I’m always in a good mood when I feel good.
I feel great today.
I’m in a good mood.
All trees have leaves.
Roses have leaves.
Roses are trees.
Pigs can’t fly.
Wilbur is a pig.
Wilbur can’t fly.
Pigs can’t fly.
Tweety can fly.
Tweety isn’t a pig.
Every adult is eligible to vote.
John is eligible to vote.
John is an adult.
Some students play football
John is a student.
John plays football
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