1. What is a set?
A set is a collection of objects.
Can you give me some examples?

2. Section 6 Concept and Notation of Sets
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)
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}

3. Elements of a Set 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.

4. Section 7 Subsets 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

5. Important subsets of R 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 half-closed interval
[a, b) = { x: x  R and a  x < b} Half-closed and half-open 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

6. Important Facts on Subsets
A  A
 A
A B and B C  A C
Can you give proofs to them?

7. Equal Sets and Proper Subsets
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

8. Section 8 Intersection and Union of Sets
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}. B A
A B

9. Union of sets A  B 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}. B A
A  B

10. Section 9 Complements 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\A B

11. Example 9.2 Ex.2.3 1-9 Given that E={1, 2, 3, 4, 5, 6, 7, 8, 9, 10},
{1, 2, 3, 4, 5, 6, 7}
{4, 5}
 (B and C are disjoint)
{1, 2, 3}
{6, 7} E
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 =
A B =
C B =
A\B =
B\A=
A  B  C =

12. Exercise (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, + )

13. 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 A 1 2 3 4 a b c d e f
Domain = A
Range = B
Codomain={a, b, c}

14. Ex.2.4, Q.6
Group discussion
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)

15. State the differences between the following functions
f: Z  Z defined by f(x) = x2
g:N  N defined by g(x)=x2

16. Injective functions
A function 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A)(x1  x2  f(x1)  f(x2))
 (x1, x2 A) (f(x1) = f(x2)  x1 = x2)

17. Examples 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

18. 2. Let a, b, c, d be real numbers and c0
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).
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.

19. 3. Strictly monotonic functions are injective.
(a) Theorem: If f : A  B is strictly increasing(or decreasing), then f is injective, where A and B are subsets of R.
Proof: For any a  b, either (i) a > b or (ii) a < b.
When a > b, f(a) > f(b)  f(a)  f(b).
When a < b, f(a) < f(b)  f(a)  f(b).
Conclusively, a  b  f(a)  f(b) and hence f(x) is injective.

20. (b) Corollary of the theorem
Corollary : If the derivative of a real-valued function f of real variables is always strictly greater(less) than 0, then f is injective.
Proof : If f (x) > 0, then f is strictly increasing and hence injective by the theorem.

21. Example Define a function f : R+  R by f(x) = x3.
Prove that f is injective.
Proof 1: Since f (x) = 3x2 > 0, f is strictly increasing and hence injective.
Proof 2:
a3 = b3 implies that (a – b)(a2 + ab + b2) = 0.
i.e. a = b or a2 + ab + b2 = 0.
However, a2 + ab + b2 = (a –b/2)2 + 3b2/4 > 0.
 a = b and thus f is injective.

22. 4. 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:
f(0)= f(0z1+0z2) = 0f(z1)+0f(z2) = 0
() 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.

23. Which of the following functions are injective? Give proofs.
Ex. 2.4 Q.10
Which of the following functions are injective? Give proofs.
g(x) = x2 + 1
f(x) = x/(1-x)
h(x) = (x + 1)/(x – 1)
k(x) = x3 + 9x2 +27x + 4

24. 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

25. Surjective Functions
A function 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)

26. (bB)(aA)(f(a) = b)? 5 y f
Examples
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

27. For any y (0, 1], then there exists x=((1-y)/y) R
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

28. 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

29. Example 1 A A B B a b c d
f -1 1 2 3 4 1 2 3 4 f a b c d

30. Example 2 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
Sketch them.

31. Example 3 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
Sketch them.

32. Example 4 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
Sketch them.

33. Graphs of a function & its inversey
y=x
y=f(x)
y=f-1(x) x

34. Composite functions of f(x)and f-1(x)
Ex.2.4 Q.11 Ex