LECTURE 1
Sets, logic &
algebra review
Lecture 1 Topics:
§
§
§
§
Set theory
A bit of logic
Mathematical proofs
Algebra review
F
F
F
F
Numbers
Rules of algebra
Inequalities
Intervals & absolute values
§ Reading:
Sydsæter et al 5th ed Ch 1, Ch 2.1 to 2.7
Sydsæter et al 4th ed Ch 3.4 to 3.7, Ch 1
2
Set Theory (1.1)
A set is a collection of elements or members. The set is
defined by listing all its members.
The elements of a set may be listed using braces { }.
Example: the students enrolled in an economics degree.
𝑆 = {𝑒𝑐𝑜𝑛𝑜𝑚𝑖𝑐𝑠 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠}
sÎS
𝑎 ∉𝑆
B is a subset of S if every element of B belong also to S.
Example: female students in economics.
𝐵 = 𝑓𝑒𝑚𝑎𝑙𝑒 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 𝑜𝑓 𝑒𝑐𝑜𝑛𝑜𝑚𝑖𝑐𝑠
BÌS
e.g.
N= {1, 2, 3, 4…..} set of natural numbers
3
Notation
§ aÎA
a is an element of set A
§ bÏA
b is not an element of set A
§ AÌB
A is a proper subset of B if all the elements
of A are contained in B but B has at least one
element not in A.
§ AÍB
A is a (weak) subset of B, all the elements
of A are contained in B but it may also be the case
that A and B are equal. (not commonly used)
§ { } or Ø a set with no elements is the empty set or
null set
4
More notation
§ sets may be described by their defining property
e.g. {x: x Ì N and x > 7}
F read "the set of x such that x is a natural number and x is
greater than 7"
§ A Ç B the intersection of two sets A and B: elements
that belong to both A and B
or {x:xÎA and xÎB}
§ disjoint sets have no elements
in common (mutually exclusive)
i.e. AÇB=Æ
5
§ A È B The union of two sets A and B: contains
elements that belong to at least one of the sets
A or B or both
B
B
{x: xÎA or xÎB}
A
§ A\B
A minus B:
the elements that belong
to A but not B
A Ç B'
B
A
6
§ universal set W
§ complement of A is all the elements that are not in A
notation A' , CA
or Ω\A or
!
A
or A
A'
A
Venn diagrams become unmanageable with more than
3 sets.
7
Set Operations (summary) (1.1)
Notation
Name
Set definition
AÈ B
A Union B
The elements belong
to A or B
AÇ B
A intersection B
Elements must belong
to both A and B
A minus B
Elements belong to A
but not to B
A\ B
Another way of defining these operations is as follow:
A È B = {x : x Î A or x Î B}
A Ç B = {x : x Î A and x Î B}
A \ B = {x : x Î A and x Ï B}
8
Example 1
§ Let A={2,3,4}, B={2,5,6}, C={5,6,2}, D={6}. Which of the
following statements is true?
4 Î C, 5 Î C
A Ì B, D Ì C
B = C,
A=B
— Find the following sets:
A Ç B,
AÈ B
A \ B, B \ A
( A È B) \ ( A Ç B)
9
Example 2
Let sets of UQ students be identified as
F = female students; B = biology students;
E = economics students; T = tennis players;
describe in words:
1.
2.
3.
4.
W\T
FÇB
BÈE
F\(EÈB)
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Other important notions about sets: Convex sets
A set is convex if a for every pair of points within the
set, every point on the straight line segment that joins
the pair of points is also within the set.
Convex sets
Non-Convex sets
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A few aspects of logic (1.2)
Solve the following equation:
"Solution":
( x + 2 )2 = ( 4 - x )
2
x+2 = 4- x
After checking, we see that
the “solution” is incorrect.
x2 + 4x + 4 = 4 - x
x2 + 5x = 0
x+5 = 0
x = -5
Some points of logical
reasoning are useful at this
time.
If x = -5
LHS = x + 2 = -3
RHS = 4 - x = 3
x+2 ¹ 4- x
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Proposition
Propositions are usually represented with capital letters: P,
Q, etc. and can be true or false
Examples: P = “all individuals who breathe are alive”; (T)
Q = “all individuals who breathe are healthy”. (F)
§ Implication arrow: P Þ Q
P implies Q
x is a square Þ x is a rectangle; x > 2 Þ x2 > 4
§ Equivalence arrow: P Û Q
P and Q are logically
equivalent
if P Þ Q and Q Þ P then P Û Q
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Necessary and sufficient conditions
PÞQ
F P implies Q, or ‘if P then Q’, or ‘Q if P’ or ‘P only if Q’
F P is a sufficient condition for Q
F also Q is a necessary condition for P
A sufficient condition for x to be a rectangle is that x is a square.
A necessary condition for x to be a square is that x is a rectangle.
PÛQ
F P is equivalent to Q , or ‘P iff Q’
F P is a necessary and sufficient condition for Q
Solving equations involves a sequence of equivalences and
implications.
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Necessary and sufficient conditions
A necessary condition is a prerequisite.
Suppose statement P is true only if another statement Q is
true; then Q is a necessary condition for P.
P Þ Q reads: “P only if Q”; “P implies Q”; “if P then Q”;
“Q if P”.
It may be that P Þ W at the same time. Then both Q and
W are necessary conditions.
Example: Let P be the statement “a person is an aunt” and Q be
the statement “a person is female”, then P Þ Q .
A person can be an aunt only if she is a female, to be a female is
a necessary condition to be an aunt. The converse is not true.
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Necessary and sufficient conditions
A sufficient condition occurs when statement Q is true if
P is true, but Q can also be true when P is not true.
The truth of P is sufficient to establish the truth of Q,
but is not a necessary condition for Q.
Given P Þ Q , then P is a sufficient condition for Q.
“P implies Q”; “if P then Q”; “Q if P”
Example: Let P be the statement “one takes a plane to
Europe” and Q be the statement “one can travel to
Europe”, then P is a sufficient condition for Q.
But sea travel is also available so we cannot say Q Þ P .
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Necessary and sufficient conditions
A condition Q can be both necessary and sufficient for
P.
P Û Q which reads “P if and only if Q” or “P iff Q”
P implies Q but Q also implies P.
Example: Let P be the statement “there are less than 30
days in the month” and Q “the month is February”, then
P ÛQ.
P only if Q, conversely Q only if P.
Q is a necessary and sufficient condition for P.
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Mathematical proofs (1.3)
In mathematics results are stated as
Theorems – the most important results, and
Lemmas – subsidiary results that support
theorems
Theorems are formulated as implications:
PÞQ
where P is a series of propositions, the premises,
and Q the conclusions.
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Mathematical proofs (1.3)
1. direct proof or deduction:
F every statement follows logically from the previous one
F PÞQ
2. indirect proof or by contradiction:
F assume proposition false
F derive implications from this and show they contradict
some proven fact
F not Q Þ not P
3. mathematical induction
F Argue from the specific to the general
F Not accepted as formal proofs
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Example 3
if –x2 + 5x – 4 > 0 prove that x must be a positive value
by indirect proof,
§ assume x £ 0
§ then 5x £ 0
§ and –x2 + 5x – 4 is the sum of three negative terms so is
less than 0 and so contradicts the proposition
§ hence solution for x must be a positive value
by direct proof
Suppose
- x2 + 5x - 4 > 0
(add x 2 + 4)
Û 5x > x2 + 4
x 2 + 4 ³ 4 for all x
Þ 5x > 4
4
Û x>
5
20
Proofs by mathematical induction (1.4)
We have a statement of the form:
Proposition: A(n) holds for all natural numbers “n”
Proof:
1. Check that A(1) is true
2. Assume that A(k) is true
3. Check that A(k+1) is true
4. If so, then A(n) is true for all natural numbers n.
21
Example 4
Prove by induction that the sum of n integers is
n(n + 1)
Sn = å x =
2
1
n
Check that it is true for n = 1 (or 2 or 3)
Assume true for n = k or Sk= ½k(k+1)
Then show that it is also true for n = k+1
S k +1 = 1 2 k ( k + 1) + ( k + 1)
=
k (k + 1) 2(k + 1) (k + 1)(k + 2)
+
=
2
2
2
i.e. true for (k+1)
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Numbers
Natural
Numbers “N ”
Integers
“Z ”
Fractions
Irrationals
“I ”
Rationals
“Q ”
Real Numbers
“R”
Complex/Imaginary
Numbers
“C ”
-1
Numbers
-5
!
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Algebra Review: Real Numbers (2.1)
N
§ Natural numbers: 1, 2, 3, 4, ...
F Odd numbers: 1, 3, 5, 7, ...
F Even numbers: 2, 4, 6, 8, ... (multiples of 2).
§ Integer numbers: ..., -3, -2, -1, 0, 1, 2, 3, .... (a more
compact notation is: 0, ±1, ± 2, ± 3, ...
Z
The Number Line
-¥ …
-3
-2
-1
0
1
2
3
…
+¥
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Algebra Review: Real Numbers
§ Rational Numbers: the ratio of two integer
numbers. Q
F Let’s call a and b two generic integer numbers;
F The ratio a/b is a rational number.
F b must be different from zero: a/0 is not defined.
§ Rational numbers can be represented using the
decimal system:
1/2 = 0.5 (finite decimal fraction)
1/3 = 0.33333... (infinite decimal fraction, or recurring or periodic)
F Every rational number can be represented as a decimal
fraction which is periodic after a finite number of digits:
11/70 = 0.1 571428 571428 571428...
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Algebra Review: Real Numbers
§ Rational numbers do not “fill” the number line!
1/4
0
1/3
1/2
1
F There is an infinite number of rational numbers
between two integers, but the number line is still
almost “empty”.
26
Algebra Review: Irrational & Real Numbers
Irrational numbers: I an infinite decimal fraction (nonperiodical) is an irrational number.
a
Irrationals: Cannot be expressed as
where a, b are integers.
b
2 = 1.414213562373....
The set of rational numbers complemented with the set of
irrational numbers give rise to the set of Real Numbers. R
Real Numbers fill the numbers line.
— The square root of a negative real is not a real number (we
need complex numbers... but not in this course).
—
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Algebra Review: Real Numbers
Textbook says the proof of 2 ¹ a b is due to Euclid
(300BC). Legend says it was proved by a Pythagorean
(probably Hippasus of Metapontum)
1
1
2
Hippasus’ proof was a proof by contradiction.
28
How the Greek’s did maths
5
4
3
Pythagoras's theorem
a 2 + b 2 = c2
29
The legend (from wikipedia):
2¹a
b
The first proof of the existence of irrational
numbers is usually attributed to a Pythagorean
(possibly Hippasus of Metapontum), who
probably discovered them while identifying
sides of the pentagram. The then-current
Pythagorean method would have claimed that
there must be some sufficiently small,
indivisible unit that could fit evenly into one of
these lengths as well as the other. However,
Hippasus, in the 5th century BC, was able to
deduce that there was in fact no common unit
of measure, and that the assertion of such an
existence was in fact a contradiction.
Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea,
and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the
universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their
ratios.” Another legend states that Hippasus was merely exiled for this revelation. Whatever the consequence to
Hippasus himself, his discovery posed a very serious problem to Pythagorean mathematics, since it shattered the
assumption that number and geometry were inseparable–a foundation of their theory.
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Closure
A set has closure under an operation if performance of that
operation on members of the set always produces a member
of the same set.
§ The set of natural numbers is closed with respect to sum.
§ Integer numbers are closed with respect to multiplication but
not division.
§ Rational numbers are a closed set with respect to division but
are not closed with respect to square root (and other powers)
§ Real Numbers are closed with respect to: sum, subtraction,
multiplication, division and powers of positive reals
31
Algebra review: Basic Arithmetic Operations
+
/
*
32
Always result in Real Numbers
With the exception:
P
: undefined for any P ÎÂ
0
Algebra review: Integer powers (2.2)
The n-times product of a real number can be
represented as an integer power:
3 × 3 × 3 × 3 = 34
a$× !#
a × a ×!"
... × a = a
n -times
— Rules of powers:
n
a = base
n = exponent
n
an
æaö
ç ÷ = n
b
èbø
a0 = 1
a
-n
1
= n
a
00 = undefined
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Algebra review: Integer powers (2.2)
— Rules of powers:
ar a s = a r + s
ar
r -s
r -s
=
a
a
=
a
as
r r
(ab )r = ab
×
ab
×
...
×
ab
=
a
×
a
×
...
×
a
×
b
×
b
×
...
×
b
=
a
b
$!#
! !"
! $!#!
" $
!#!
"
r -times
r -times
r -times
— Wrong rules of powers:
(a + b ) ¹ a n + b n
(2 + 1)2 = 9 ¹ 22 + 12 = 5
n
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Rules of Algebra (2.3)
§ commutative law
§ a+b = b+a
ab = ba
§ associative law
§ (a+b)+c = a+(b+c)
a(bc) = (ab)c
§ distributive law
§ a(b+c) = ab + ac
(a+b)c = ac + bc
35
Rules of Algebra
a+0=a
a + (-a) = 0
1.a = a
aa -1 = 1 for a ¹ 0
(-a)b = a(- b) = - ab
(-a)(- b) = ab
36
Rules of Algebra
Some more important ones:
“quadratic identities”
(a + b )2 = a 2 + b 2 + 2ab
(a - b )2 = a 2 + b 2 - 2ab
(a - b)(a + b) = a 2 - b2
37
Fractions: Rules for Operations (2.4)

‚
ƒ
„
a.c a
=
b.c b
(b¹0, c ¹ 0)
-a ( -a )( -1) a
=
=
-b ( -b )( -1) b
a
a ( -1) a -a
- = ( -1) =
=
b
b
b
b
†
‡
ˆ
a b a +b
+ =
c c
c
b a.c + b
a+ =
c
c
‰
a c ad + bc
+ =
b d
bd
b a.b
a. =
c
c
a c
a.c
. =
b d
b.d
a
a c
÷ = b
c
b d
d
=
ad
bc
38
Algebra review: Fractional Power (2.5)
A fractional power has a rational number as
exponent. In general:
a1 n = n a
a
p q
q
= a =
p
a1 n is the exponential form,
n
( a)
q
p
a is the radical form
Note:
a = a1 2 is the square root of a and is
defined as the nonnegative number that multiplied by
itself gives a.
39
Rules for Fractional Powers
ab =
for a ³ 0,
a b
also ab = ( ab )1 2 = a1 2b1 2
a
b
a
for a ³ 0, b > 0
b
=
12
also
a æaö
=ç ÷
b èbø
a+b ¹
40
= a1 2b -1 2
a + b
b ³0
Algebra review: Inequalities (2.6)
§ A strictly positive (negative) number is written as:
a>0
( a < 0)
§ We say a is bigger than b if their difference is positive
§ We say a is smaller than b if their difference is
negative
§ A weakly positive (negative) number is written as:
a³0
a is weakly positive/non-negative
( a £ 0)
41
Î "is an
element of"
Inequalities
§ > means 'is greater than';
<
‘is less than’
§ a < b on number line a is to left
of b
§ > & < strict inequalities;
§ £ & ³ weak inequalities
§ if two sides of an inequality are
multiplied by a negative number,
the direction is reversed
§ inverting an inequality reverses
the direction
Examples
§ x>2
§ 5³x
§ 5 £ x £ 10
x Î [5, 10] closed interval
§ 5 < x < 10
x Î (5, 10) open interval
§ if x < b
-x > -b
§ if 0 < x < b then
1 1
>
x b
42
Inequalities & Sign Diagrams (2.6)
Example 5: Use a sign diagram to find when the inequality
(x - 1)(3 - x) > 0 holds.
-3
-2
-1
0
1
2
3
4
5
x -1
3-x
(x -1)(3 – x)
Answer: (x - 1)(3 - x) > 0 iff 1 < x < 3
43
Inequalities & Sign Diagrams
Example 6
Use a sign diagram determine when the following
inequality holds:
2p -3
> 3- p
p -1
44
Intervals (2.7)
Let a and b be any two numbers on the real line.
Then we call all the numbers that lie between these
two numbers an interval.
Notation
Name
All x such that:
(a,b)
Open interval
a< x<b
[a, b]
Closed interval
a£ x£b
(a,b]
Half open interval
a< x£b
[a,b)
Half open interval
a£ x<b
45
Absolute Values (2.7)
For any real number, its absolute value is defined as
the distance from the origin:
ìa if a ³ 0
a =í
î- a if a < 0
|x| < b
|x| > b
means -b < x < b
x > b and x < -b
46
Next Week …
§ Summation notation
§ Solving equations
47