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

Section 6Concept 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 q0} R={r: r is a real number} C={a+bi: a and b are real

and i2=-1}

Elements of a Set

4N 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

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)(xA xB) Can you give me some examples? N Z Q R C

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

Important Facts on Subsets

A A A A B and B C A C Can you give proofs to them?

Equal Sets and Proper Subsets

A = B iff A B and B A iff (x)(xA xB)

Let A, B be two sets. A is a proper subsets of B, denoted by A B≠

⊂

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: xA and xB}.

A BA B

Union of sets

Definition 8.2

Let A and B be sets.The union of A and B is the set A B ={x: xA or xB}.

A BA B

B\A

Section 9Complements

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 }BA

Example 9.2

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 =

{1, 2, 3, 4, 5, 6, 7}

{4, 5}

(B and C are disjoint)

{1, 2, 3}

{6, 7}

E

Ex.2.3 1-9Ex.2.3 1-9

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, + )

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.

1234

abcde

fA B

Domain = A

Range = B

Codomain={a, b, c}

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) = logx2. g:R R is defined by g(x)= x3. h:N N is defined by h(x) = x/24. p:R R is defined by p(x) = cosx5. q: [-2, 3] R is defined by q(x) = (x2 -2x – 3)

Ex.2.4, Q.6Ex.2.4, Q.6

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

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, x2A)(x1 x2 f(x1) f(x2)) (x1, x2 A) (f(x1) = f(x2) x1 = x2)

Examples

1. 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 c0. 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, x2R\{-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. 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.

(i) When a > b, f(a) > f(b) f(a) f(b).

(ii) When a < b, f(a) < f(b) f(a) f(b).

Conclusively, a b f(a) f(b) and hence f(x) is injective.

(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.

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.

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, z2C.

(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 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.

Which of the following functions are injective? Give proofs.

1. g(x) = x2 + 1

2. f(x) = x/(1-x)

3. h(x) = (x + 1)/(x – 1)

4. k(x) = x3 + 9x2 +27x + 4

Ex. 2.4 Q.10Ex. 2.4 Q.10

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

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,

(bB)(aA)(f(a) = b)

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 = yTherefore f is surjective.

??

5y

f

Group Discussion on Ex.2.4 Q.10Group Discussion on Ex.2.4 Q.10

(bB)(aA)(f(a) = b)

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.10Ex.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: BA of the function f is defined as

f-1= { (b, a) : (a, b) f }

Ex.2.4 Q.10Ex.2.4 Q.10

Example 1

1

2

3

4

a b

c

d

f

A B

1

2

3

4

a b

c

d

f -1

B A

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.

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.

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.

Graphs of a function & its inverse

y=f(x)

y=f-1(x)

x

y

y=x

Composite functions of f(x)and f-1(x)

f(f-1(x))=f-1(f (x))=

X

X

Ex.2.4 Q.11

Ex.2.5 1-3

Ex.2.4 Q.11

Ex.2.5 1-3