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. Let bn be given by a recurrence relation: b1 = 1, b2 = 2
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 and a counterexample for properties which do not hold.
T
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) sets R, S , and T such that R S , S T , and R T .
/
(c)
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 assume that m and n are odd numbers, and that m2 + n2 = q 2
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. (16 points) Let S = cfw_2, 3, 2, cfw_1, 2, 3, 4 . For ea
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, explain why.
(a) A B .
(b) A set X A such that |X | = 1.
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, explain why.
(a) A B .
Note that the only element A and
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).
(c) (P Q) (P Q)
2. (12 points) For each natural number
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 and a counterexample for properties which do not hold.
T
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 tautology.
P
T
T
F
F
Q
T
F
T
F
P Q
T
F
F
F
(P Q) P
T
T
T
T
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
T
F
T
F
P
F
F
T
T
QP
T
T
F
T
Q
T
F
T
F
P Q
T
F
F
F
(P )
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 placing all element in U in
appropriate region. Then p
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 positive integer n, it is the case that 1 + 5 + 9 + + (4n
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 doesnt exist.
(a) A set A of 3 elements, such that A P (S )
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 might be curious about which elements of B are actually a
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, explain why.
(a) A B .
Note that the only element A and
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 something specic but not overspecic: noting that, for
in
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 3x +2 = 0, then it follows from basic
algebraic methods
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 following statement to be true: For integers a and b,
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 nonconstructive proof ) or prove them
false (by a disproof of exis
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 , (a, a) R;
/
in other words, a R a for all a S .
(a) (2
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 of 3 (i.e., removing a number
of objects which is a powe
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, 10, . . ..
4x if x > 0
.
2 4x if x 0
It is quite easy
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 every element of A is an element of B and C , every elemen
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 such a set doesnt exist.
(a) A set A of 4 elements such
k EA,
311'spring 2016 Jan. 16.
(print)
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4.
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