National University of Singapore
Department of Mathematics
GEK1505 Living with Mathematics
Tutorial 3
1. Find the total number of (i) squares, (ii) rectangles that can be formed
from the gridlines in an 8 9 chequered board (Figure 1).
(Think of the best w

Selected slides on Graphs
For the rest of the material in the
lectures, please refer to the
textbook Living with Mathematics,
McGraw-Hill.
3. Graphs
3.1 The seven bridges of Konigsberg
In 1736, the following problem was posed to the
Swiss mathematician Le

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Counting 3
For the rest of the material in the
lectures, please refer to the
textbook Living with Mathematics,
McGraw-Hill.
Given n different objects, in how many ways can you
choose k of them?
This is equivalent to asking:

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Reasoning 1
For the rest of the material in the
lectures, please refer to the
textbook Living with Mathematics,
McGraw-Hill.
The art of reasoning consists in getting hold
of the subject at the right end, seizing on the
few g

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Reasoning 2
For the rest of the material in the
lectures, please refer to the
textbook Living with Mathematics,
McGraw-Hill.
Foundations of modern logic laid by
Bertrand Russell (1872 - 1970)
Alfred North Whitehead (1861 1

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Reasoning 3
For the rest of the material in the
lectures, please refer to the
textbook Living with Mathematics,
McGraw-Hill.
Some commonly encountered examples :
1. If it rains, I will stay at home.
2. If you finish your hom

Selected slides from lectures:
Counting 2
For the rest of the material in the
lectures, please refer to the
textbook Living with Mathematics,
McGraw-Hill.
2.3 Review of Sets
Questions like What is the population of
Timbuktu?, How many eggs are in the
bask

Selected slides from lectures:
Counting 1
For the rest of the material in the
lectures, please refer to the
textbook Living with Mathematics,
McGraw-Hill.
Generalized Tower of Hanoi Problem
If in the Tower of Hanoi Problem, there
are n slabs to start with

National University of Singapore
Department of Mathematics
GEK1505 Living with Mathematics
Tutorial 2
1. Suppose that the following statement is a true statement: If I have said
it once before, I have said it 100 times. How many times have I said it?
2. F

National University of Singapore
Department of Mathematics
GEK1505 Living with Mathematics
Tutorial 6
1. Assume that a person shakes hands with another person at most once and
that a handshake between two persons is counted as one handshake.
(a) Draw a gr

Selected slides from lectures:
Reasoning 4
For the rest of the material in the
lectures, please refer to the
textbook Living with Mathematics,
McGraw-Hill.
The mode or argument or reasoning takes the form:
pq
(1)
qr
(2)
Therefore, p r
(3)
This is called t

National University of Singapore
Department of Mathematics
GEK1505 Living with Mathematics
Tutorial 5
1. Among 3 given (not necessarily distinct) positive integers, which of the
following statements is true and which is not necessarily true? Justify
your

Tutorial 11: Solutions
Q. 1
B
A
C = A'
D = D'
E = B'
F = C'
(a) Clockwise rotation about D through 90o.
Q 1 (b)
B
E =B
=D'
C =A'
G=A'
P
A
D =D'
F=C=C'
H=B'
(b) (Clockwise) rotation about P through 180o.
Q 1 (c)
O
B
C
E=D'
G=A' =C'
I =D'
A
D
F=C'
H=B' =B'