MA 1100 (SEMESTER I 2015/2016) MIDTERM TEST 1 SOLUTION
DATE : 17 SEPTEMBER 2015
1.
Show that for any natural number n N , there exists a natural number m N such that
42n+1 + 3n+2 = 13 m.
[Write your proof within this page]
[15 points]
Solution. For each n
MA1100
Lecture 8
Mathematical Proofs
Proof by Contradiction
Existence Proof
Chartrand: section 5.2, 5.4
When using cases, they must cover all possibilities
Fr
From last lecture
A universal statement involves two real numbers x
and y.
Which of the followin
MA1100
MA1100
Lecture 8
Mathematical Proofs
Proof by Contradiction
Existence Proof
Chartrand: section 5.2, 5.4
Announcement
Homework
Lecture Quiz
HW2 due on September 16
Error in HW2 Q5b
HW1 scores uploaded to gradebook
LQ2 scores uploaded to gradebook
Mi
MA1100
MA1100
Lecture 7
Mathematical Proofs
Basic properties of numbers
Congruences
Proof by Cases
Chartrand: section 3.2, 3.3, 4.1
1
True Statements
A proposition is a true mathematical statement
that has a proof.
Chartrand calls it Result
A propositio
MA1100
Lecture 7
Mathematical Proofs
Basic properties of numbers
Congruences
Proof by Cases
Chartrand: section 3.2 - 3.4, 4.1-4.2
1
Announcement
Homework set 2 available
Due on September 16
Cover topics from last Friday to next Tuesdays
lectures
Within mi
MA1100
MA1100
Lecture 6
Logic
Quantified statements
Mathematical Proofs
Parity & Divisibility
Direct Proofs
Proof by Contrapositive
by Co
Chartrand: section 2.10, 3.2, 4.1
1
Announcement
Homework set 1 due today
Hand in during lecture break or by end of
t
MA1100
Lecture 5
Logic
Logical equivalence
Quantified statements
Chartrand: section 2.8 2.10
1
Negation (revisited)
Example (daily life)
John is tall and thin
Negation:
PQ
~(P Q)
John is not tall and thin
John is not tall and not thin
John is not tall or
MA1100
Lecture 5
Logic
Logical equivalence
Quantified statements
Chartrand: section 2.8 2.10
1
Announcement
Homework set 1 due next week (Friday)
Available in Workbin:
Homework problem set 1
Hints for HW1
Marking scheme and HW format
Next Tuesday (Hari Ra
MA1100
Lecture 4
Logic
Implication
Converse and contrapositive
Biconditional statements
Logical equivalence
Chartrand: section 2.4 2.8
Implication
Standard form: If P then Q
Symbolic form: P Q
Also called a conditional statement
P is called the hypothesis
MA1100
MA1100
Lecture 4
Logic
Implication
Converse and contrapositive
Biconditional statements
Logical equivalence
Chartrand: section 2.4 2.8
Announcement
Homework set 1 due next week (Friday)
Homework problem sheet in workbin
Read the instruction on the
MA1100
MA1100
Lecture 9
Mathematical Proofs
Proving involving
Sets relations
Power sets
Cartesian products
Indexed collection of sets
Empty sets
Chartrand: section 4.4, 4.5, 4.6
1
every element of A is also an element of B
Proving A B
Example
A, B are sub
MA1100
MA1100
Lecture 10
Mathematical Induction
Axiom of Induction
Principle of Mathematical Induction
Base Case
Inductive Step
Chartrand: section 6.1, 6.2
Announcement
Term break next week
Week 7 (week after term break)
no lecture & tutorial
revision web
Unions
Distributivity
DeMorgans Laws
Boolean Algebra
The Axiom of Unions
Bernd Schroder
Bernd Schroder
The Axiom of Unions
logo1
Louisiana Tech University, College of Engineering and Science
Unions
Distributivity
DeMorgans Laws
Boolean Algebra
The Axiom o
A/U/T*
Student Number:
*Delete where necessary
NATIONAL UNIVERSITY OF SINGAPORE
FACULTY OF SCIENCE
SEMESTER 1 EXAMINATION 2012-2013
MA1100
Fundamental Concepts of Mathematics
November/December 2012
Time allowed: 2 hours
INSTRUCTIONS TO CANDIDATES
1. Write
MA 1100
NATIONAL UNIVERSITY OF SINGAPORE
MA 1100 Fundamental Concepts of Mathematics
(Semester 1 : AY2015/2016)
Final Examination 30 November 2015
Time allowed : 2 hours
Student ID Number:
A
INSTRUCTIONS TO STUDENTS
(1) This test paper contains TEN (10) q
MA 1100
NATIONAL UNIVERSITY OF SINGAPORE
MA 1100 Fundamental Concepts of Mathematics
(Semester 1 : AY2014/2015)
Final Examination 1 December 2014
Time allowed : 2 hours
Student ID Number:
INSTRUCTIONS TO STUDENTS
(1) This test paper contains TEN (10) ques
MA1100
Student Number:
NATIONAL UNIVERSITY OF SINGAPORE
MA1100 - Fundamental Concepts of Mathematics
(Semester 1 : AY2013/2014)
Name of examiner : Assoc Prof Tan Victor
Time allowed : 2 hours
INSTRUCTIONS TO CANDIDATES
1. Write down your matriculation/stu
Properties Sets Should Have
The Problem
The Remedy
Russells Paradox
Bernd Schroder
Bernd Schroder
Russells Paradox
logo1
Louisiana Tech University, College of Engineering and Science
Properties Sets Should Have
The Problem
The Remedy
If We Could Define Se
Fundamental Concepts of
Mathematics
Lecture 01
Natural Numbers
Summation and Product Notation,
Mathematical Induction,
AM-GM Inequality
Chin CheeWhye
Department of Mathematics
National University of Singapore
AY 2016/2017 Semester 1
Natural Numbers N
N :=
Definition
Examples
Truth Values
Statements
Bernd Schroder
Bernd Schroder
Statements
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Examples
Truth Values
A statement or proposition is a sentence that is
either true or false
MA1100
Lecture 10
Mathematical Induction
Axiom of Induction
Principle of Mathematical Induction
Base Case
Inductive Step
Chartrand: section 6.1, 6.2
Example 1
Cardinality of power set
Let A be a finite set with n elements. How many
elements does P(A) have
MA1100
Lecture 3
Sets
Partitions of Sets
Cartesian Products of Sets
Logic
Statements
Logical operations
Chartrand: section 1.5 1.6, 2.1 2.3
Indexed Collection of Sets
1
Cn = x R | x n
n
What are
C
C
n
n
and
for all n N
C
n
n
?
n
n
C
n
n
MA1100
Lecture 3
2
MA1100
MA1100
Lecture 2
Sets
Set Relations
Set Operations
Indexed Collection of Sets
Chartrand: section 1.2 1.4
Subsets
Let A be the set cfw_ x U | P(x) with universal set U.
We say A is a subset of U and write A U.
Let A and B be two sets with universal
MA1100 Lecture 19
Functions/Number Theory
Inverse functions Greatest Common Divisor Euclidean Algorithm
1
Common divisor
Recall 1 Let n be an integer and m a non-zero integer. We say m divides n and write m | n if there exists an integer q such that n =
MA1100 Lecture 18
Functions
Composition of functions Inverse functions
1
Composition of functions
Example f: R R defined by f(x) = 3x2 + 2 g: R R defined by g(x) = sin(x) g(f(x)= g( 2 + 2) = sin(3x2 + 2) 3x f g R R R
f( g(x)= f(sin(x) = 3sin(x)2 + 2 g f
MA1100 Lecture 17
Functions
Injection Surjection Bijection
1
Announcement
Homework 4 due next Tuesday Homework 3 scores in Gradebook Mid-term test scores will be in Gradebook by today. Students who missed tutorial class this week can pick up their test