1. Sequence A list of objects arranged in a particular order.
Order is very important in a sequence. (Unlike Sets).
Finite sequence=> No. of elements is finite.
Example: 1, 4, 7, 10, 13.
Mathematically, a n = a n-1 + 3, where a1 = 1 and 1≤n<6
Infinite=> No. of elements is not finite.
Example: 1, 4, 7, 10, 13, …
Three dots means “and so on”.
Mathematically, a n = a n-1 + 3, where a1 = 1 and 1≤n<∞

2. Sequence
Recursive: If we need previous element to get the next element in a sequence, then the formula is called recursive.
Example: 1, 4, 7, 10, 13, …
Three dots means “and so on”.
Mathematically, a n = a n-1 + 3, where a1 = 1 and 1≤n<∞
Explicit: Without using the previous element we will be able to get the exact value of each element in a sequence. This type of formula is called explicit.
Example: 1, 4, 9, 16, 25, …
Mathematically, b n = n 2, where 1≤n<∞

3. Sequence What is the explicit formula for the following sequence?
-2, 4, -8, 16, -32, …
Explicit formula:
b n = (-2) n, where 1≤n<∞
What is the recursive formula for the following sequence?
1, 4, 7, 10, 13, …
Recursive formula:
b n = b n-1 + 3, where b 1 = 1 and 1≤n<∞

4. Set and Sequence
The set corresponding to a sequence is simply the set of all distinct elements in the sequence.
Example1:
Sequence: -2, 4, -8, 16, -32, …
The set corresponding to the above sequence:
{-2, 4, -8, 16, -32, …}
Example2:
Sequence: 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1
{0, 1}

5. Characteristic Functions
If A is a subset of a universal set U, the characteristic function f A of A is defined for each x Є U as follows:
f A (x) = 1 if x Є A
= 0 if x  A
Use:
Help us prove theorems about properties of subsets
To represent sets in a computer.

6. Properties of Characteristic Functions
f A∩B = f A f B
=> f A∩B (x) = f A(x) f B(x) for all x
f AUB = f A + f B - f A f B
=> f AUB(x) = f A(x) + f B(x) - f A(x) f B(x) for all
f AB = f A + f B - 2f A f B
=> f AB(x) = f A(x) + f B(x) - 2f A(x) f B(x) for all x

7. Proof of the properties of characteristic functions
f A∩B = f A f B
=> f A∩B (x) = f A(x) f B(x) for all x
RHS= f A(x) f B(x)
= 1 if and only if f A(x) = 1and f B(x) =1
= 1 if and only if x Є A and x Є B
= 1 if and only if x Є A∩B
= f A∩B (x) (since f A f B is 1 on A∩B and 0 otherwise)
= LHS

8. Computer representation of sets and subsets
If a universal set U is finite and A is a subset of U, then the characteristic function assigns 1 to an element that belongs to A and 0 to an element that does not belong to A.
Thus fA can be represented by a sequence of 0’s and 1’s of length n.
Example:
U = {1,2,3,4,5,6}
A = {1,2}, B = {2,4,6}, and C = {4,5,6}
fA corresponds to the sequence 1,1,0,0,0,0.
fB corresponds to the sequence 0,1,0,1,0,1.
fC corresponds to the sequence 0,0,0,1,1,1.

9. Computer representation of sets and subsets
Example:
U = {a, b, c, d, e, f, g, h}
A = {a, b, c, d}
fA(x) = 1 if x = a, b, c, d
= 0 if x = e, f, g, h 1 U A 1

10. Countable and uncountable sets
Countable set => A set is called countable if it corresponds to some sequence. Members of such a set can be arranged in a list. Members can be counted.
Example: All finite sets are countable.
Uncountable set => A set that is not countable is called uncountable.
Example: Set of all real numbers that can be represented by an infinite decimal of the form
0.a1a2a3…where ai is an integer and 0≤ai ≤ 9

11. 1.3 Exercises Give three different sequences that have
{1, 2, 3, …} as a corresponding set.
Possible answers:
1,2,3,4,…
1,1,2,2,3,3,4,4,…
1,2,2,2,3,2,4,2,5,2

12. 1.3 Exercises Write the formula for the nth term. 0, 3, 8, 15, 24, 35
b n = n 2 - 1, where 1≤n ≤ 6 => Explicit
0, 2, 0, 2, 0, 2, …
b n = 0, if n is odd
= 2, if n is even => Explicit
2, 5, 7, 12, 19, 31,…
b n = b n-1 + b n-2 , b 1 = 2, b 2 = 5 => Recursive