Name: ______
MAT 101 – Survey of Mathematical Reasoning April 10, 2014
Professor Pestieau Exam 2 – Propositional Logic
Multiple-Choice Questions [5 pts each]
Circle the correct answer on the following multiple-choice questions.
1. Which of the following is not a statement?
a) “Some counting numbers are odd but not prime.”
b) “The number is extremely large.”
c) “Some triangles have 4 sides if and only squares have exactly 5 corners.”
d) “For some integer , .”
2. Only one of the statements below is not true. Which one is it?
a) “Every set has a subset that is empty.”
b) “Two consecutive integers can be odd.”
c) “Suffolk County is located to the East of Nassau County.”
d) “Two lines that cross each other can be perpendicular but not parallel.”
3. The negation of the statement “Nothing can be moving all the time” is given by
a) “Something can be moving all the time.”
b) “Something can be moving sometimes.”
c) “Something cannot be moving all the time.”
d) “Something cannot be moving sometimes.”
4. If is true and , are false component statements, then what is the truth value of the compound statement ?
a) True. b) False. c) Neither.
For questions 5 and 6, let represent the statement “She has long hair”, represent the statement “He has a brown beard,” and represent the statement “I remember this couple.”
5. Translating the symbolic statement verbally yields
a) “Neither does she have long hair nor does he have a brown beard.”
b) “She does not have long hair or he does not have a brown beard.”
c) “Either she does not have long hair or he does not have a brown beard.”
d) Any one of the choices above… they are all logically equivalent.
6. Writing the verbal statement “If she has long hair and he does not have a brown beard, then I do not remember this couple” symbolically yields
a)
b)
c)
7. What can you conclude, in general, about the symbolic statement , where are component statements?
a) It is a tautology (i.e. it is always true).
b) It is a self-contradiction (i.e. it is always false).
c) It is only false when both and are false.
d) It is neither a contradiction nor a tautology.
8. How many rows would be needed to construct the truth table of a compound statement with 6 component statements?
a) 8 b) 16
c) 32 d) 64
Show all your work on the following problems to receive full credit.
Problem 1 [15 pts]
Write the following quotations symbolically using logical symbols. Specify the letters you attribute to each of the component statements in the quotation explicitly. Do not forget to remove negations from these components.
a) “That’s one small step for man, but one giant leap for mankind.”
- Neil Armstrong
b) “If you don’t stop and look around for a while, then you could miss it.”
- Ferris Bueller
c) “Ready or not, here I come.”
- The Delfonics
Problem 2 [10 pts]
Write the arithmetical statement “” symbolically as a conjunction of two disjunctions. Specify the letters you attribute to each of the component statements you use explicitly.
Problem 3 [15 pts]
Consider the following statement:
“Everyone on Long Island was affected by super storm Sandy in 2012, but some did not lose power in their homes.”
a) Write this statement symbolically. Specify the letters you attribute to each of the components explicitly.
b) Use De Morgan’s law to write the negation of this statement verbally.
Problem 4 [25 pts]
a) Using the table below, show that the symbolic statement Had a pretty tough day today. is a tautology.
p / q / rT / T / T
T / T / F
T / F / T
T / F / F
F / T / T
F / T / F
F / F / T
F / F / F
b) Consider the following three component statements:
p – “All triangles have more than 3 sides.”
q – “An odd number has a factor of 2.”
r – “The state of New York in located on the West Coast of the U.S.A.”
Show that the statement iis still true with these components. Justify your answer using the table you constructed in part a).
c) Draw the circuit that corresponds to the statement . Explain why he statement is a tautology using this circuit.
Bonus Problem [5 pts]
What is the truth table of the compound statement “Neither p nor q,” where p and q are component statements?