MAD 2104 Midterm: MAD 2104 FIU Exam 111k
MAD 2104 Sept 1, 2011
Quiz I Prof. S. Hudson
1) Prove that ¬p↔qis logically equivalent to p↔ ¬qusing a table and a few words (you
can use another proof style, for partial credit, at your own risk).
2a) Find the negation of ∀x(x2> x) (answer without using ¬).
2b) The proposition in 2a) is true for some domains and not others. Give an example of
a domain for which it is true. Your answer should be a set of numbers.
2c) Give a domain for which it is false.
3) Answer each part with “True” or “False”. You don’t have to explain (but it doesn’t
hurt, and might help if we decide later that a question was not totally clear).
a) p∨(p→q) is a tautology.
b) (p→q)∧(p→ ¬q) is a contradiction.
c) The converse of p→ ¬qis q→ ¬p.
d) p→ ¬qis logically equivalent to q→ ¬p.
e) 0101 L1100 = 1101.
Bonus (approx 5 pts): Show that {∧,¬} is functionally complete. You can quote any HW
problems you did related to this. Hint: Find an equivalence that shows ∨can be expressed
with {∧,¬} (and explain your reasoning). You can answer on the back, with a note.
Remarks and Answers: I wanted this Quiz to be a pleasant welcome to the course, but
probably made it a bit too easy; the average was 89 out of 100, based on the top 30 scores.
The unofficial scale is;
A’s 90 - 100
B’s 80 - 89
C’s 70 - 79
D’s 60 - 69
F’s 50 - 59
Normally, the average grade will be a C+, more-or-less. Probably, I should raise this
scale a bit higher, but it is already as high as any I’ve ever used. Answers are below:
1
Document Summary
Prof. s. hudson: prove that p q is logically equivalent to p q using a table and a few words (you can use another proof style, for partial credit, at your own risk). 2a) find the negation of x(x2 > x) (answer without using ). 2b) the proposition in 2a) is true for some domains and not others. Give an example of a domain for which it is true. Your answer should be a set of numbers. 2c) give a domain for which it is false: answer each part with true or false . Bonus (approx 5 pts): show that { , } is functionally complete. You can quote any hw problems you did related to this. Hint: find an equivalence that shows can be expressed with { , } (and explain your reasoning). You can answer on the back, with a note.
