Assignment 1 ue by Midnight Nov. 29th.
Plagiarism:
University policy on
Academic honesty is critical to the integrity and quality of any
(online) degree program and enforced by all faculty in the BSCS
program. Please review the university definition of academic
dishonesty and refer to your course outline for possibly additional
policies in this regard.
Academic Dishonesty (from NU catalog):
Academic dishonesty includes cheating, plagiarism, and any attempt
to obtain
Credit for academic work through fraudulent, deceptive, or
dishonest means.
Below is a list of some forms academic dishonesty may take.
sing or attempting to use unauthorized materials,
information, or study Aids in any academic exercise.
ubmitting work previously submitted in another course
without the consent of the instructor.
itting for an examination by surrogate or acting as a
surrogate.
epresenting the words, ideas, or work of another as oneàown
in any Academic exercise.
onducting any act that defrauds the academic process.
1. [10 points] Write a truth table for each expression [hint: see section 1.2 and 1.3 under
additional problems]
a. (r ? p) ? ( ? i
b. (p ? q) ? (q ? )
2. [6 points] Give an English sentence in the form of “If…then….” that is equivalent to each
sentence. [hint: see section 1.3 under additional problems]
a. Maintaining a B average is sufficient for Joe to be eligible for the honors
program.
b. Maintaining a B average is necessary for Joe to be eligible for the honors
program.
c. Rajiv can go on the roller coaster only if he is at least four feet tall.
3. [4 points] Give the converse, contrapositive, and inverse of the following propositions:
a. If it snows tomorrow, John will not drive.
Converse:
Contrapositive:
Inverse:
b. If Jill does not arrive within 5 minutes, she will miss the flight.
Converse:
Contrapositive:
Inverse:
4. [4 points] Use truth table to investigate whether the pair of expressions are logically
equivalent: 0 ? i and ? q [hint: see section 1.4 under additional problems]
5. [10 points] Prove or disprove equivalence: (p ? r) ? (q ? r) ? (p ? q) ? r [hint: see
section 1.5 under additional problems]
a. Use truth table
b. Use laws of propositional logic
6. [4 points] In the following question, the domain of discourse is a set of students at a
university. Define the following predicates: [hint: see section 1.7 under additional
problems]
•
E(x): x is enrolled in the class
T(x): x took the test
Translate the following English statements into a logical expression with the same
meaning.
a.
b.
c.
d.
Someone who is enrolled in the class took the test.
All students enrolled in the class took the test.
Everyone who took the test is enrolled in the class.
At least one student who is enrolled in the class did not take the test.
7. [8 points] Use De Morgan’s law for quantified statements and the laws of propositional
logic to show the following equivalences: [hint: see section 1.8 under additional
problems]
ø ((x) ? (Q(x) ? ¨x))) ? ?x (P(x) ? (hx) ? R(x)))
8. [4 points] Prove that the argument is valid using a truth table. [hint: see section 1.11
under additional problems]
p?q
p?q
?p
9. [4 points] Prove that the argument is valid using rules of inferences. [hint: see section
1.12 under additional problems]
p?q
q?r
Š?
10. [4 points] Prove that the following argument is valid by replacing each proposition with
a variable to obtain the form of the argument. Then use the rules of inference to prove
that the form is valid. [hint: see section 1.12 under additional problems]
If I work out hard, then I am sore.
If I am sore, I take an aspirin.
I did not take an aspirin.
? I did not work out hard.
11. [2 points] Find a counterexample to show that each of the statements is false. For every
positive integer x, x3 < 2x.
12. [6 points] Give direct proof: If x, y, and z are integers such that x|(y+z) and x|y, then
x|z. [ [hint: see section 2.4 under additional problems]
13. [6 points] Prove by contrapositive: For every integer n, if 5n + 3 is even, then n is odd.
[hint: see section 2.5 under additional problems]
14. [4 points] Consider the sets A= { 2, 4, 5, 8, 10, 12} and B = { 1, 3, 10, 9}. Find
?????=
?????=
?????=
?????=
15. [4 points] Use the set identities to prove the following identity. [hint: see section 3.5
under additional problems]
????????
?=??
????
?????

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