Sequences
Sequences (examples)
•
•
•
•
1,2,3,4,5,6,
1,3,5,7,9,11,13
2,4,6,8,10,12,14
1,1,2,3,5,8,13
Sequences
• Characteristics
– Arranged in a definite order
– Finite or Infinite
– Elements may be distinct, or repeated
• Example:
– S,U,P,E,R,M,A,N = distinct elements
– Number 15 in binary representation = 001111
» 0’s as first and second elements, 1’s as third, fourth, fifth
and sixth elements
– 1,3,5,7,9,… = infinite distinct elements ( “…” means and so on)
Try these
• Predict the next element of the following
sequences
– 1, 3, 5, 7, 9, …
– 2, 4, 6, 8, 10, 12, …..
– 1, 1, 2, 3, 5, 8, 13, ….
– 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
Sequences
• 2 kinds
– Recursive - Refers to previous terms to define the
next terms. Every recursive formula must include a
starting place
– Explicit - Refers to position of elements in the
sequence. For example, the nth element will be
referred as an
Sequences: Recursive formula
Write recursive formula for the following
sequences:
Hint:**previous term, next term & starting point
1. 2, 6, 10, 14, 18, 22, …
2. 3, 9, 15, 21, 27, …
3. 5, 1, -3, -7, -11
Sequences: Explicit formula
• Write explicit formula for the following
sequences:
• 2, 4, 8, 16, 32, …
• 1, 4, 9, 16, 25, …
• -3, 1, 5, 9, …
Sequences
• May also contain alphabets or words
– Example:
• “INTERNET” = a finite sequence of an English word
– I,N,T,E,R,N,E,T
• “TODAY IS A BEAUTIFUL DAY” = sequence of words,
called strings
– “TODAY, IS, A, BEUATIFUL, DAY”
Set Corresponding to a Sequence
• Refers to the set of all distinct elements in the
sequence
– Example: 1,0,0,1,0,1,0,0,1,1,1
• Set corresponding = {1,0} or {0,1}
• Try these
– B,A,N,A,N,A
– D,I,S,C,R,E,T,E,S,T,R,U,C,T,U,R,E
– S,E,H,A,T,I,S,E,J,I,W,A
Characteristic Functions
• A function on a set that assigns some ‘value’ to
each element of the set
• If A is a subset of a universal set U, the
characteristic function, fA is defined as:
1 if x  A
fA (x) =
0 if x  A
• We may add and multiply characteristic
functions, since their values are numbers
Theorems…
1. f A ∩ B = fAfB ; that is
f A ∩ B (x) = fA(x)fB(x) for all x
2. f A ᴜ B = fA + fB - fAfB ; that is
f AᴜB (x) = fA(x) + fB(x) - fA(x)fB(x) for all x
3. f A⊕B = fA + fB - 2 fAfB ; that
f A⊕B (x) = fA(x) + fB(x) - 2 fA(x)fB(x) for all x
Computer representation of sets &
subsets
• To represent a set in a computer, elements of
the sets must be arranged in a sequence
• When a universal set is finite, U = {x1, x2, x3, …,
xn}, then characteristics function assigns 1 for
xi that belongs to A and 0 for xi that does not
belong to A.
• Thus, fA can be represented by 0’s and 1’s of
length n.
Computer representation of sets &
subsets
• Example:
– U = {1,2,3,4,5,6,7,8,9}
– A = {2,3,5}
fA =
0
1
1
0
1
0
0
0
0
– B = {1,3,5,7}
fB =
1
0
1
0
1
0
1
0
0
– C = {8}
fC =
0
0
0
0
0
0
0
1
0
Try these…
• Recursive formula
– Write the recursive formula for the following
sequence
• 2, 4, 6, 8, 10
• 3, 9, 27, 81
• 2, 5, 26, 677
Try these…
• Recursive formula
– Write the first four terms for the following formula
• a1 = 4, an = an-1 + 5
• a1 = -3, an = (an-1)3
Try these…
• Explicit formula
– Write the recursive formula for the following
sequence
• 2, 4, 6, 8, 10
• 3, 9, 27, 81
• 4, 24, 124, 624