A Set in Mathematics is a Collection of Well Defined and Distinct Objects
Part B Set Theory What is a set? A set is a collection of objects. Can you give me some examples?
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Transcript of Part B Set Theory What is a set? A set is a collection of objects. Can you give me some examples?
Part B Set Theory
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, x2)(x1 x2 f(x1) f(x2)) (x1, x2) (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. 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
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
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
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