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 p
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 i
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 up
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 topi
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
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
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 ca
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 i
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 (we
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
Di
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
Ti
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
INSTRUCTI
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
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
INSTRUCT
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 S
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 S
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
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 finit
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
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
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
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
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