MA121 STUDY GUIDE WEEK 1
1. Set-builder notation, union, and intersection
Reading assignment: Sections 1.1 1.4
After the lecture, you should be able to do the following.
Understand set-builder notation, and be able to describe the elements of
a (reasonab
Winter Term, 2014
Name: : I Cl (AND/3
Section: __
WILFRID LAURIER UNIVERSITY
Waterloo, Ontario
Mathematics 121 Introduction to Mathematical Proofs
Test 2 March 21, 2014
Instructors:
S. Bauman (Section B - 8:30 am TR)
F. Vinette (Section C 5:30
Example 1. Let a, b Z and n N. Prove that a b (mod n) if and only
if b a (mod n).
Solution. Suppose that a b (mod n). Then there exists k Z such
that a b = kn. Thus b a = (a b) = (kn) = (k)n. And since
k Z whenever k Z it follows that n divides b a. This
MA121 Quiz: January 31, 2013.
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provide justification for your answer. Please use the back of this page if additional space
MA121 Midterm Test Winter 2013
[10 marks]
Page 1 of 7
1. Let A = cfw_1, 2, 3 and B = cfw_1, 3.
(a) List all elements of A B.
Solution:
A B = cfw_(1, 1), (1, 3), (2, 1), (2, 3), (3, 1), (3, 3).
(b) List five elements of P(A).
Solution:
P(A) = cfw_, cfw_1,
MA121B Quiz Solutions: March 7, 2013.
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provide justification for your answer. Please use the back of this page if additiona
MA121C Quiz: March 7, 2013.
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back of this page if additional space is needed.
1. Let p be the statement: there exists a r
MA121C Quiz: April 4, 2013.
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1. (a) Demonstrate how the Euclidean Algori
MA121C Quiz: March 21, 2013.
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1. Let z, w C. Prove that if |z| = |w| = 1
Example 1. Let a, b Z. Prove that if b|a then any prime factor of b is
also a prime factor of a.
Solution. If b|a then there exists k Z such that a = b k. Now let A denote
the set of distinct prime factors of a, B the set of distinct prime factors of
b an
MA121A Quiz: February 7, 2013.
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provide justification for your answer. Please use the back of this page if additional space
MA121 Quiz: January 17, 2013
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1. State which of the following sentences are statements. If a sentence is a statement, determine its negation.
(a) x2 + 2x +
MA121B Quiz: March 21, 2013.
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1. Prove or disprove the following stateme
MA121B Quiz: April 4, 2013.
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1. Suppose that a, b and c are integers. Pr
MA121 Mock Test 2
Name:
* Please remember that mock tests are meant as a means of providing an extra set of practice questions
and basis for a review class. Do not study for the midterm based solely on the topics covered by the mock
test! Go back through
Fundamental Theorem of Arithmetic and Divisibility Review Mini Lecture
Here we will provide a proof of the Fundamental Theorem of Arithmetic (about prime factorizations).
Before we get to that, please permit me to review and summarize some divisibility fa
MA121 Mock Test 1
Answers
(Full Solutions will NOT be posted;
use the MACs drop-in help centre if you have any questions.)
* Please remember that the mock test was meant as a means of providing an extra set of practice
questions and basis for a review cla
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1. Two statements p and q are equivalent if:
-1
p q is a tautology.
2
They have the same truth table.
3
They use the same connectives.
5
They have the same disjunctive normal form.
7
Neither of them is a tautology.
j_j
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j_j
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When is an a
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If the number 3 is even then the number 4 is even.
If the sun rises from the west, then a rooster
lays eggs.
The sun rises from the west if and only if a
rooster lays eggs.
Which of the following sets are equal to A = cfw_2, 4, 6, 8?
-1
B is the set of single digit, even, natural numbers.
2
C = cfw_6, 8, 2, 4
3
D = cfw_2, 4, 4, 6, 8
5
E = cfw_2, 4, 6, 8,
7
F = cfw_x N : x even and 2 x 10
U
U
Charmi Desai
MA 121
Sept 8, 2016
Chapter 1: Sets
o Set is defined by:
1. Give a precise verbal definition of the set stating exactly what its elements are
2. Make a complete list of all elements in the set
Ex. cfw_?,?,?,?
3. Make a list of few members of
Example 1. Use a Venn Diagram to evaluate/simplify each of the following
(a) (A \ B) (A B).
(b) (A \ B) B.
(c) (A # B) (A B).
Solution. See the following three pages.
1
Example 2. Prove that
(a) A = (A \ B) (A B).
(b) A B = (A \ B) B.
(c) A B = (A # B) (A
Example 1. Let A = cfw_a, b. Determine P(P(A).
Solution. In class we observed that (the list given in class involved a lot of
. . . and was therefore incomplete)
P(P(A) = cfw_, cfw_B, cfw_C, cfw_D, cfw_E, cfw_B, C, cfw_B, D, cfw_B, E, cfw_C, D,
cfw_C, E,
Example 1. Let z, w 2 C. Prove that z + w = z + w.
Solution. Using the definitions of conjugatation and addition we have
Re(z + w) = Re(z + w)
= Re(z) + Re(w)
= Re(z) + Re(w)
And
Im(z + w) =
Im(z + w)
=
[Im(z) + Im(w)]
=
Im(z)
=
Im(z) + [ Im(w)]
Im(w)
= I
Example 1. Suppose that z1 , z2 and z3 are complex numbers. Prove that
z1 (z2 + z3 ) = z1 z2 + z1 z3 .
Proof. We must show that
Re(z1 (z2 + z3 ) = Re(z1 z2 + z1 z3 )
(1)
Im(z1 (z2 + z3 ) = Im(z1 z2 + z1 z3 ) .
(2)
and
In order to do so we will put each nu
WILFRID LAURIER UNIVERSITY
WATERLOO, ONTARIO
Sessio-n: Winter Term, 2010
Nance:
Course no: MA121
ID
#:
Title: Introduction to
Mathematical Proofs
Section:
Instructors: F. Vinette, (Section B): R. Cressman, (Section C)
Number of Pages: 9. plus cover page
L
MA121 Mock Final Exam
Name:
* Please remember that mock tests are meant as a means of providing an extra set of practice questions
and basis for a review class. Do not study for the midterm based solely on the topics covered by the mock
test! Go back thro