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

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}

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.

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

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)(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 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

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

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

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 =

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

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)

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, x2A)(x1  x2  f(x1)  f(x2))  (x1, x2 A) (f(x1) = f(x2)  x1 = x2)

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

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.

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.

(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, 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.

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

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, (bB)(aA)(f(a) = b)

(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

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

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

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

Composite functions of f(x)and f-1(x) Ex.2.4 Q.11 Ex.2.5 1-3