MATH 311-02
Notes
Introduction to Higher Math
Even when we dont have an explicit formula for a recurrence, we can often work out valuable
information about it using inductive methods:
Proposition 1. L
MATH 311-01
Midterm #2
1. (18 points) For each of the following relations, determine whether it is reexive, symmetric,
and/or transitive, providing a brief explanation for the properties which hold an
MATH 311-01
Practice Midterm Examination
Name:
1. (12 points) Give examples of sets satisfying the following conditions, or explain why they
cannot be met:
(a) sets A, B , and C such that A B
C.
(b) s
MATH 311-01
Practice Midterm Solutions
1. (14 points) Prove that if m and n are odd numbers, then m2 + n2 is not a perfect square.
(Hint: what is the parity of m2 + n2 ?)
Let us counterfactually assum
MATH 311-01
Final Examination
1. (12 points) Identify each of the following statements as a tautology, a contradiction, or neither.
Show your work.
(a) (P Q) P .
(b) (P Q) (P Q).
(c) (P Q) (P Q).
2. (
MATH 311-01
Practice Final Examination
1. (15 points) Let A = cfw_, 3, 4, cfw_10, 12 and let B = cfw_3, 4, cfw_10. Find sets matching the
following descriptions or, where such a set does not exist, ex
MATH 311-02
Final Examination Solutions
1. (15 points) Let A = cfw_, 3, 4, cfw_10, 12 and let B = cfw_3, 4, cfw_10. Find sets matching the
following descriptions or, where such a set does not exist, e
MATH 311-01
Midterm Examination
Name:
1. (12 points) Write out truth tables for each of the following statements (you may write them
all in one truth table, if you wish):
(a) (P ) (Q P ).
(b) P (P Q).
MATH 311-01
Midterm #2
1. (18 points) For each of the following relations, determine whether it is reexive, symmetric,
and/or transitive, providing a brief explanation for the properties which hold an
MATH 311-01
Final Examination Solutions
1. (12 points) Identify each of the following statements as a tautology, a contradiction, or neither.
Show your work.
(a) (P Q) P .
As seen below, this is a tau
MATH 311-01
Midterm Examination
Name:
1. (12 points) Write out truth tables for each of the following statements (you may write them
all in one truth table, if you wish):
(a) (P ) (Q P ).
P
T
T
F
F
Q
Test 1 VIAT 311 pring 2&6.
~{AME \ u . ' (print)
1. (lllptzs) Let U {1,2, - - ,9 10} be the universal set and let A {2,3,4 5,6} and B
{5, 6: 7, 8} be two SLllJbClb‘. Draw a Venn diagram for A and B by
MATH 311-01
Practice Midterm #2
1. (14 points) Prove that if m and n are odd numbers, then m2 + n2 is not a perfect square.
(Hint: what is the parity of m2 + n2 ?)
2. (12 points) Prove that for any po
MATH 311-01
Practice Midterm Solutions
1. (12 points) Let S = cfw_0, 1, 2, 3. For each of the following descriptions, either produce a set
matching the description or explain briey why such a set does
MATH 311-02
1
Notes
Introduction to Higher Math
Function Attributes and Properties
If a function is dened as f : A B , then the set A is called the domain of f ; the set B is the
codomain of f .
We mi
MATH 311-02
Final Examination Solutions
1. (15 points) Let A = cfw_, 3, 4, cfw_10, 12 and let B = cfw_3, 4, cfw_10. Find sets matching the
following descriptions or, where such a set does not exist, e
MATH 311-02
Problem Set #1 Solutions
1. (10 points) We observed in class that when A is a nite set, |P (A)| = 2|A| . Explain in your
own words why this is true.
We might be best served by looking at s
MATH 311-02
Problem Set #2
1. (10 points) Consider the statements P : x2 3x + 2 = 0 and Q : x 0.
(a) (6 points) Explain in words why P Q is true.
If we were to presume that x satises the equation x2 3
MATH 311-02
Problem Set #3
1. (15 points) Below are two proofs relating to divisibility of products.
(a) (7 points) Determine the sucient (and if possible necessary) conditions on an integer
n for the
MATH 311-02
Problem Set #4 Solutions
1. (12 points) Below are three existence statements which are either true or false. For each of
them, either prove them true (by either an example or a nonconstruc
MATH 311-02
Problem Set #5
1. (9 points) The following questions will explore this slightly obscure relation property.
Denition 1. A relation R on a set S is antireexive if and only if, for all a S ,
MATH 311-02
Problem Set #6 Solutions
1. (6 points) In power-of-3 nim, the game state is a single non-negative integer (i.e., a stack of
coins), such that each move consists of subtracting some power o
MATH 311-02
Problem Set #6
1. (24 points) Demonstrate the existence of bijections between the following pairs of sets.
(a) (4 points) The set Z and the set of positive even integers E = cfw_2, 4, 6, 8
MATH 311
Midterm Examination
1. (12 points) Give examples of sets satisfying the following conditions, or explain why they
cannot be met:
(a) sets A, B , and C such that A B C .
Any sets such that eve
MATH 311-02
Final Examination Solutions
1. (12 points) Let S = cfw_2, 3, 2, cfw_1, 2, 3, 4 . For each of the following descriptions, either
produce a set matching the description or explain briey why
k EA,
311'spring 2016 Jan. 16.
(print)
3 each of the following sets in the form {:I; e Z : \vhere is the property
1, 3,—5, }
{,x: Ix:_(2n.—§)-1Q,Y- Wgsw I’IZ-I}
2,4,8, ~-}
{2" r 1420 t}: WSW}
4.
{1&3