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Part B Set Theory
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What is a set?
A set is a collection of objects.
Can you give me some examples?
Section 6
Concept and Notation of Sets
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Tabular Form
N={1, 2, 3, 4,…}
Z={0, -1, 1, 2, -2,…}
Q=?
R=?
C=?
S={1, 2, 3, 4}
T={fish, fly, a, 4}
={ } ( is called the
empty set)
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Set-Builder Form
N={n: n is a natural number}
Z={m: m is an integer}
Q={p/q: p and q are
integers and q0}
R={r: r is a real number}
C={a+bi: a and b are real
and i2=-1}
Elements of a Set
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4N means that:
4 is an element of N;
4 is a member of N;
4 belongs to N;
4 is contained in N;
N contains 4.
Section 7 Subsets
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Definition 7.1
Let A and B be two sets. A is a subset of B iff
every element of A is an element of B.
Symbolically, A  B iff (x)(xA xB)
Can you give me some examples?
NZQRC
Important subsets of R
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Let a, b be two real numbers with a  b
(a, b) = { x: x  R and a < x < b} Open interval
[a, b] = { x: x  R and a  x  b} Closed interval
(a, b] = { x: x  R and a < x  b} Half-open and halfclosed interval
[a, b) = { x: x  R and a  x < b} Half-closed and halfopen interval
(a, +) = {x : x  R and x > a}
[a, +) = {x : x  R and x  a}
(- , a) = {x : x  R and x < a}
(- , a] = {x : x  R and x  a}
(- , +) = R
Important Facts on Subsets
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AA
 A
A B and B C  A C
Can you give proofs to them?
Equal Sets and Proper Subsets
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A = B iff A  B and B  A
iff (x)(xA  xB)
Let A, B be two sets. A is a proper subsets of B,
denoted by A ⊂
B
≠
Section 8
Intersection and Union of Sets
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Definition 8.1
Let A and B be sets.The intersection of A and B
is the set A B ={x: xA and xB}.
A
A B
B
Union of sets
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Definition 8.2
Let A and B be sets.The union of A and B is the
set A  B ={x: xA or xB}.
A
ABB
Section 9
Complements
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Definition 9.1,2
Let A and B be sets. The complement of A in B
is defined as the set
B\A={x: x B and x A }
A
B
B\A
Ex.2.3 1-9
Example 9.2
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Given that
E={1, 2, 3, 4, 5, 6, 7, 8, 9, 10},
A = {1, 2, 3, 4, 5}, B = { 4, 5, 6, 7} and C = { 8, 9, 10}
A  B = {1, 2, 3, 4, 5, 6, 7}
A B = {4, 5}
C B =  (B and C are disjoint)
A\B = {1, 2, 3}
B\A= {6, 7}
ABC= E
Exercise
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(1, 5)  (3, 8)
(1, 5)  (3, 8)
(-10, 1]  [1, 4]
(-10, 1]  [1, 4]
(-, 3)  (-1, +)
(-, 3) (7, 100)
R\Q
R\(1, 5)
(1,5 )\(3, 7)
(3, 8)\[2, 9]
(5, +)\(1, 3]
=(3, 5)
=(1. 8)
={1}
=(-10, 4]
=R
=
=Set of all irrational numbers
=(-, 1] [5, +)
=(1, 3]
=
=(5, + )
Section 10 Functions
Definition
f: A B is a function from a set A to a set B
iff f assigns every object in A a unique image in
B.
B
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A
Domain = A
Range = B
Codomain={a, b, c}
f
1
2
3
4
a
b
c
d
e
Ex.2.4, Q.6
Group discussion
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Refer to Ex.2.4 Q.5, discuss on which are graphs of
functions and state their domains, ranges and
codomains.
Determine which of the following are functions:
1. f: R R is defined by f(x) = logx
2. g:R  R is defined by g(x)= x
3. h:N  N is defined by h(x) = x/2
4. p:R  R is defined by p(x) = cosx
5. q: [-2, 3]  R is defined by q(x) = (x2 -2x – 3)
State the differences between the
following functions
 f:
Z  Z defined by f(x) = x2
 g:N
 N defined by g(x)=x2
Injective functions
f: A  B is called an
injection (injective function or
one-to-one function)
 iff it doesn’t assign two distinct
objects to the same image.
Symbolically,
 (x1, x2)(x1  x2  f(x1)  f(x2))
  (x1, x2) (f(x1) = f(x2)  x1 = x2)
 A function
Examples
1.
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Is the function f: N N defined by f(x) =
2x injective?
How to prove it?
Proof:
f(x1) = f(x2)
 2x1 = 2x2
 x1 = x2
 f is injective
2. Let a, b, c, d be real numbers
and c0. f: R\{-d/c}R be a function
defined by f(x)=(ax+b)/(cx+d).
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Show that if ad-bc 0, then f is injective.
Proof:
Let x1, x2R\{-d/c}, and suppose that f(x1)=f(x2),
then (ax1+b)/(cx1+d)= (ax2+b)/(cx2+d)
 (ad-bc)(x1-x2) = 0

x1=x2 (Since ad-bc 0)
 f is injective.
3. Let f:C C be a function satisfying
f(az1+bz2)=af(z1)+bf(z2) for any real
numbers a and b and any z1, z2C.
(a) Show that f(0) = 0
(b) f is injective iff when f(z)=0 we have z=0.
Proof:
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f(0)= f(0z1+0z2) = 0f(z1)+0f(z2) = 0
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Proof:
() when f(z)=0, then f(0)=0=f(z)  z=0 since f is injective.
() If f(z1) = f(z2), then f(z1) - f(z2)= 0
 f(z1-z2) = 0
 z1-z2 = 0
 z1 = z2 . Thus f is injective.
Ex. 2.4 Q.10
Which of the following functions
are injective? Give proofs.
1.
2.
3.
4.
g(x) = x2 + 1
f(x) = x/(1-x)
h(x) = (x + 1)/(x – 1)
k(x) = x3 + 9x2 +27x + 4
State the difference between the
following functions
 h: Z
 Z defined by h(x) = x + 1
and
 k: N  N defined by k(x) = x + 1
Surjective Functions
f: A  B is called an
surjection (surjective function or
onto function) iff
 every element of B is an image of an
element in A. Symbolically,
 (bB)(aA)(f(a) = b)
 A function
(bB)(aA)(f(a)
Examples
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= b)
?
?
f
5
y
Prove that f: R R defined by
f(x) = 3x + 2 is surjective.
Proof: For any real number y, there
exists a real number x = (y – 2)/3 such
that
f(x) = 3((y – 2)/3) + 2 = y
Therefore f is surjective.
Group Discussion on Ex.2.4 Q.10
2. Show that the function f: R(0, 1]
defined by f(x) = 1/(x2+1) is surjective.
 Proof:
For any y (0, 1], then there exists
x=((1-y)/y) R
such that f(x)=1/((1-y)/y+1)=y.
Therefore f is surjective.
Ex.2.4 Q.10
Bijective Functions and their
inverse functions
 Let
f: A B be a funcition. f is called
a bijective function(or bijection) iff f is
both injective and surjective.
 The inverse function f-1: BA of the
function f is defined as
f-1= { (b, a) : (a, b)  f }
Ex.2.4 Q.10
Example 1
A
B
A
B
f -1
f
1
a
a
1
2
b
b
2
3
c
c
3
4
d
d
4
Example 2
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Let f: R R be a function defined by
f(x) = 2x –1.
Then f is bijective.
Since y = 2x –1  x = (y + 1)/2
f-1(x) = (x + 1)/2
Example 3
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Let f: R+ R be a function defined by
f(x) = log10x
Then f is bijective.
Since y = log10x  x = 10y
f-1(x) = 10x
Example 4
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Let f: [0, +) [0, +) be a function defined by
f(x) = x2
Then f is bijective.
Since y = x2  x = +y,
f-1(x) = +x
Graphs of a function & its inverse
y
y=f(x)
y=x
y=f-1(x)
x
Composite functions of
f(x)and f-1(x)
 f(f-1(x))= X
 f-1(f (x))=X
Ex.2.4 Q.11
Ex.2.5 1-3