Download HW-04 due 02/10

CmSc180 Discrete Mathematics
Homework 04 due 02/10
1. For the implication ~P → Q indicate which of the following expressions is its
contrapositive, its converse and its inverse (underline the correct one):
Contrapositive:
P → Q, ~Q → P,
P → ~Q,
~P → ~Q,
~Q → ~P, Q → ~P
Inverse:
P → Q, ~Q → P,
P → ~Q,
~P → ~Q,
~Q → ~P, Q → ~P
Converse:
P → Q, ~Q → P,
P → ~Q,
~P → ~Q,
~Q → ~P, Q → ~P
2. Let P , Q, and R be the propositions
P: I am awake
Q: I work hard
R: I dream of home
Represent each of the following sentences as logical expressions:
a. I dream of home only if I am not working hard
b. Working hard is sufficient for me not to dream of home
c. Being awake is necessary for me to work hard
3. Give direct proof of the following statement:
For all n, if n is even then (n-1)(n+1) is odd
4. Prove by mathematical induction
12 + 22 + 32 + 42 + 52 + …. + n 2 = n(n+1)(2n+1)/6
5. Using the predicates class(x), difficult (x), boring(x) and appropriate quantifiers
(,), represent in predicate logic the following sentences, write the negation of
the predicate expression and translate back to English
a. Some classes are difficult and boring.
b. Difficult classes are not boring.
c. No classes are difficult and boring
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6. 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 (converse or inverse error).
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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