Math 310
Exam 1 - Practice Problems
1. Translate the following English statements into symbolic form (use propositional sentences).
(a) I will not pass this class unless I go to class every day and do all of the homework exercises.
(b) I lock the doors an
Math 310: Discrete Mathematics
Some Important Logical Equivalences
Equivalence
pTp
pFp
pT T
pF F
ppp
ppp
( p) p
pq qp
pq qp
(p q ) r p (q r )
(p q ) r p (q r )
p (q r ) (p q ) ( p r )
p (q r ) (p q ) ( p r )
( p q ) p q
( p q ) p q
p (p q ) p
p (p q )
Math 310
Quiz 1
01/13/2014
Name:
1. Charles Lutwidge Dodgson was a writer, mathematician, logician.
(a) Give the name of the city where Charles Lutwidge Dodgson was born and the year he was born.
(b) Briey describe one of Charles Lutwidge Dodgsons contrib
Math 310
Solutions to Selected Problems from HW #6
2.1 #16 Find two sets such that A B and A B .
Solution:
First, we should note that to have any chance of A being both an element of B and a subset of B , the
set B that we choose must contain at least one
Math 310
Solutions to Selected Problems from HW 2
Section 1.2 #41: This question asks us to construct a logical expression involving p, q , and r which is true when exactly
two of these variables are true and false otherwise. The hint suggests using a dis
Math 310
Solutions to HW 1.1
Problem #45: Each inhabitant of a remote village always tells the truth or always lies. A villager will only give a Yes
or a No response to a question a tourist asks. Suppose you are a tourist visiting this area and come to a
Math 310
Exam 2
Name:
Instructions: You will have 55 minutes to complete this exam. The credit given on each problem will be
proportional to the amount of correct work shown. Answers without supporting work will receive little credit.
1. (8 points) Let k
Math 310
Gateway Quiz 2 - Version 1
Name:
1. (8 points) Suppose that r is a non-zero rational number. Use the denition of rational to prove that rk is
rational for any k Z.
First, apologies for the minor typo in the original statement of this problem. The
Math 310
Exam 1 - Version 1
Name:
Instructions: You will have 55 minutes to complete this exam. The credit given on each problem will
be proportional to the amount of correct work shown. Answers without supporting work will receive little
credit.
1. (10 p
Math 310
Gateway Quiz 1 - Version 1
Solutions
Name:
1. (8 points) Write the negation of the following symbolic statement. Your answer should be written
in fully simplied form (i.e. so that no negation is outside a quantier and any negation symbols
should
Math 310
Exam 2 - Practice Problems
1. Prove that there is an integer m such that m3 > 1010 . Is your proof constructive or non-constructive?
2. Prove that given any two rational numbers p < q , there is a rational number r with p < r < q .
3. Prove that
Math 310
Exam 2 - Practice Problem Solutions
1. Prove that there is an integer m such that m3 > 1010 . Is your proof constructive or non-constructive?
Let m = 2200. Then m3 = (2200)3 = 10, 648, 000, 000 > 1010 = 10, 000, 000, 000.
This proof is constructi
Math 310
Exam 1 - Practice Problem Solutions
1. Translate the following English statements into symbolic form (use propositional sentences).
(a) I will not pass this class unless I go to class every day and do all of the homework exercises.
First we dene
Math 310
Logic and Truth Tables
Denition: A proposition is a declarative sentence that is either true or false (but not both). That is, a statement ends
up having one of two possible truth values:
If a statement is true, we assign it the truth value T .