Functions and Algorithms - McMaster Universityms.mcmaster.ca/~nahid/Math-301/Lecturenotes/... · Exponential and logarithmic functions Sequence and Summation Sequence Summation Indexed

OutlineFunctions, Domain, Codomain, Range

One-to-one and Onto functionComposition function and Inverse function

Mathematical functionSequence and Summation

Recursively defined functionsAlgorithms and Function

Cardinality

Functions and Algorithms

Math 301

Dr. Nahid Sultana

October 5, 2012

Math 301 Functions and Algorithms





Cardinality

Functions, Domain, Codomain, Range

One-to-one and Onto function

Composition function and Inverse function

Mathematical functionFloor and ceiling functionInteger remainderExponential and logarithmic functions

Sequence and SummationSequenceSummationIndexed collection of sets

Recursively defined functions

Algorithms and Function

Cardinality






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I Definition: A function is a rule that for every input assign aspecific output.

I Think function as a machine in which each input producesexactly one output.

I Suppose A and B are two sets. A function f : A→ B is a rulethat for every element a ∈ A assign a single value f (a) from B.

I A is called the domain of the function f .

I B is called the codomain of the function f .

I f (a) is called the image of a.

I the set of all image values is called the range of f .






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I Functions as relations: A function f : A→ B is a relationfrom A to B such that each a ∈ A belongs to unique orderedpair (a, b) in f .

I Example: A = {1, 2, 3} and B = {x , y , z ,w}

f1 = {(1, y), (2,w), (3, y)}

f2 = {(1, x), (2, x), (3, y), (2, z)}

f3 = {(1, z), (3, x)}

I Example: f (x) = 3x − 4.

I Two functions f : A→ B and g : A→ B will be equal iff (a) = g(a) for all a ∈ A.






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f1 = {(1, y), (2,w), (3, y)}

f2 = {(1, x), (2, x), (3, y), (2, z)}

f3 = {(1, z), (3, x)}








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f1 = {(1, y), (2,w), (3, y)}

f2 = {(1, x), (2, x), (3, y), (2, z)}

f3 = {(1, z), (3, x)}








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f1 = {(1, y), (2,w), (3, y)}

f2 = {(1, x), (2, x), (3, y), (2, z)}

f3 = {(1, z), (3, x)}








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I A function is said to be one-to-one (injective) if differentelements in the domain have distinct images.

I A function f : A→ B is said to be one-to-one iff (a1) = f (a2) implies a1 = a2.

I Graphical Example.I Example: f (x) = x2 and f (x) = x3.I Example: f (x) = 3x − 5 is one-to-one.I Onto: Each element of the codomain is the output/image of

some element of the domain.I A function f : A→ B is onto (surjective) if the range of f is

the entire codomain. i.e. if f (A) = B.I The function f is called one-to-one correspondence (or

bijective) if it is both one-to-one and onto.I Graphical Examples.






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I Graphical Example.I Example: f (x) = x2 and f (x) = x3.

I Example: f (x) = 3x − 5 is one-to-one.I Onto: Each element of the codomain is the output/image of









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I Graphical Example.I Example: f (x) = x2 and f (x) = x3.I Example: f (x) = 3x − 5 is one-to-one.

I Onto: Each element of the codomain is the output/image ofsome element of the domain.

I A function f : A→ B is onto (surjective) if the range of f isthe entire codomain. i.e. if f (A) = B.

I The function f is called one-to-one correspondence (orbijective) if it is both one-to-one and onto.

I Graphical Examples.






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some element of the domain.

I A function f : A→ B is onto (surjective) if the range of f isthe entire codomain. i.e. if f (A) = B.








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the entire codomain. i.e. if f (A) = B.








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bijective) if it is both one-to-one and onto.







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I A composition function can be thought of as combination oftwo functions.

I The composition of two given function f and g is denoted byf ◦ g and written as

(f ◦ g)(x) = f (g(x))

I Example: f (x) = 2x − 1 and g(x) = x3 − 5, Find (f ◦ g)(3).






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(f ◦ g)(x) = f (g(x))







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(f ◦ g)(x) = f (g(x))







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I Suppose f (x) and g(x) are two bijective functions. If(f ◦ g)(x) = x and (g ◦ f )(x) = x , then we say that f (x) andg(x) are inverse of each other. i.e

f (x) = g−1(x) and g(x) = f −1(x)

I A function that has an inverse is called invertible.

I Theorem: A function f : A→ B is invertible iff f is abijective function.

I Notes:I f(x) must be bijectiveI f −1(x) does not mean 1

f (x)

I Domain of f −1 = Range of fI Range of f −1 = Domain of f






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f (x) = g−1(x) and g(x) = f −1(x)




f (x)







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f (x) = g−1(x) and g(x) = f −1(x)




f (x)







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f (x) = g−1(x) and g(x) = f −1(x)




f (x)







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I Example: Is f (x) = x2 is invertible.

I Example: Find the inverse function of f (x) = 3x − 7.






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I Example: Is f (x) = x2 is invertible.

I Example: Find the inverse function of f (x) = 3x − 7.






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Floor and ceiling functionInteger remainderExponential and logarithmic functions

I Definition: The floor of a real number x is the biggestinteger which is smaller or equal to x , is denoted by bxc.

I Example: 3.14, -8.5

I Definition: The ceiling of a real number x is the smallestinteger which is greater or equal to x , is denoted by dxe.

I Example: 3.14, -8.5

I Definition: The integer value of a real number x converts xinto an integer by deleting the fractional part of the number,Written as INT(x).

I Example: 3.14, -8.5, 7

I The absolute value of a real number x , written as ABS(x) |x |I Example: -15, 7, -3.33






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I Example: 3.14, -8.5


I Example: 3.14, -8.5


I Example: 3.14, -8.5, 7







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I Example: 3.14, -8.5


I Example: 3.14, -8.5


I Example: 3.14, -8.5, 7







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I Example: 3.14, -8.5


I Example: 3.14, -8.5


I Example: 3.14, -8.5, 7







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I Let k be any integer and M be a positive integer then

k(mod M)

denote the integer remainder when k is divided by M.

I Example: 25(mod 7), 25(mod 5)

I If k is negative then k(mod M) = M − r ′, where r ′ is theremainder when |k | is divided by M.

I Example: −26(mod 7)






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k(mod M)










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k(mod M)










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k(mod M)










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I Integer exponents

a.a.a.....a︸︷︷︸n times

= an, where a and n are both positive integers

I Exponential function

f (x) = ax , where a is positive integer and x is real number

I Logarithmic function: The logarithm of any positive numberx to base b is written as

logbx

I Natural logarithm is the logarithm to the base e

logex or ln(x)

I Exponential and logarithmic functions are inverse of eachother

e ln(x) = x and ln(ex) = x if x > 0






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I Integer exponents






logbx


logex or ln(x)








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I Integer exponents






logbx


logex or ln(x)








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I Integer exponents






logbx


logex or ln(x)








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I Integer exponents






logbx


logex or ln(x)


e ln(x) = x and ln(ex) = x if x > 0Math 301 Functions and Algorithms





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SequenceSummationIndexed collection of sets

I A sequence is a discrete structure used to represent anordered list.

I A sequence is a function f : N→ A.

I A sequence is usually denoted by

a1, a2, a3, .... or {an} or {an}∞n=1

I a1, a2, a3, ...an, finite sequence

I Example: {an}, where an = 1/n.

I Example: 3, 8, 13, 18, .....

I A formula that refers to previous terms to define the nextterm is called recursive.

I Example: 1, 4, 9, 16, 25, .....






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a1, a2, a3, .... or {an} or {an}∞n=1



I Example: 3, 8, 13, 18, .....


I Example: 1, 4, 9, 16, 25, .....






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a1, a2, a3, .... or {an} or {an}∞n=1



I Example: 3, 8, 13, 18, .....


I Example: 1, 4, 9, 16, 25, .....






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a1, a2, a3, .... or {an} or {an}∞n=1



I Example: 3, 8, 13, 18, .....


I Example: 1, 4, 9, 16, 25, .....






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a1, a2, a3, .... or {an} or {an}∞n=1



I Example: 3, 8, 13, 18, .....


I Example: 1, 4, 9, 16, 25, .....






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a1, a2, a3, .... or {an} or {an}∞n=1



I Example: 3, 8, 13, 18, .....


I Example: 1, 4, 9, 16, 25, .....






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a1, a2, a3, .... or {an} or {an}∞n=1



I Example: 3, 8, 13, 18, .....


I Example: 1, 4, 9, 16, 25, .....






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I Summation:

n∑j=1

aj = a1 + a2 + a3 + ....+ an

n∑j=m

aj = am + am+1 + am+2 + ....+ an

Here j is called the dummy index of summation.

I Example: What is the value of∑5

j=1 j2.






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I Summation:

n∑j=1

aj = a1 + a2 + a3 + ....+ an

n∑j=m

aj = am + am+1 + am+2 + ....+ an

Here j is called the dummy index of summation.

I Example: What is the value of∑5

j=1 j2.






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I A non empty set I , called an index set, is used to select thesets we want to consider.

I {Sα}α∈I describe the collection of all sets Sα, called theindexed collection of sets.

I The union of the sets in {Sα}α∈I is defined as⋃α∈I

Sα = {x |x ∈ Sα for some α ∈ I}

I The intersection of the sets in {Sα}α∈I is defined as⋂α∈I

Sα = {x |x ∈ Sα for all α ∈ I}

I Example: Suppose Sn = {n, 2n}, where n is a positiveinteger. Compute

⋃α∈I Sα, where I = {1, 2, 4}.






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⋃α∈I Sα, where I = {1, 2, 4}.






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⋃α∈I Sα, where I = {1, 2, 4}.






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⋃α∈I Sα, where I = {1, 2, 4}.






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⋃α∈I Sα, where I = {1, 2, 4}.






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I Factorial function:1. If n = 0, then n! = 12. If n > 0, then n! = n.(n − 1)!

I Example: Calculate 4!.

I Fibonacci Sequence:1. If n = 0 or n = 1, then Fn = n2. If n > 1, then Fn = Fn−2 + Fn−1

I 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, .......I Ackermann function:

1. If m = 0, then A(m, n) = n + 12. If m 6= 0 but n = 0, then A(m, n) = A(m − 1, 1)3. If m 6= 0 and n 6= 0, then A(m, n) = A(m − 1,A(m, n − 1))

I Find A(1, 3).






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I Example: Calculate 4!.I Fibonacci Sequence:

1. If n = 0 or n = 1, then Fn = n2. If n > 1, then Fn = Fn−2 + Fn−1

I 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, .......

I Ackermann function:1. If m = 0, then A(m, n) = n + 12. If m 6= 0 but n = 0, then A(m, n) = A(m − 1, 1)3. If m 6= 0 and n 6= 0, then A(m, n) = A(m − 1,A(m, n − 1))

I Find A(1, 3).






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I Example: Calculate 4!.I Fibonacci Sequence:

1. If n = 0 or n = 1, then Fn = n2. If n > 1, then Fn = Fn−2 + Fn−1

I 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, .......I Ackermann function:

1. If m = 0, then A(m, n) = n + 12. If m 6= 0 but n = 0, then A(m, n) = A(m − 1, 1)3. If m 6= 0 and n 6= 0, then A(m, n) = A(m − 1,A(m, n − 1))

I Find A(1, 3).






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I An algorithm is a finite step-by-step list of well-definedinstructions for solving a particular problem.

I Example: (Polynomial evaluation)

f (x) = 2x3 − 7x2 + 4x − 15 .

Compute f (5).I There are two methods:

1. Direct method.( n(n+1)2 multiplication and n addition )

2. Synthetic division. ( n multiplication and n addition )

I Synthetic division method is more efficient than the directmethod.






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f (x) = 2x3 − 7x2 + 4x − 15 .

Compute f (5).

I There are two methods:









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f (x) = 2x3 − 7x2 + 4x − 15 .










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f (x) = 2x3 − 7x2 + 4x − 15 .










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I “Big O” notation: Suppose p(x) is a polynomial of degreem,then

p(x) = O(xm) .

I Example: 7x2 − 9x + 4 = O(x2)






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I “Big O” notation: Suppose p(x) is a polynomial of degreem,then

p(x) = O(xm) .

I Example: 7x2 − 9x + 4 = O(x2)






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I A set is finite if it has n ∈ N distinct elements.

I The cardinality of a finite set A is the number of elements inA, it is denoted by |A| or n(A) or card(A).

I Two sets A = {a, b, c} and B = {1, 2, 3} have samecardinality.

I A set A is said to be infinite if it is not finite.

I Two sets A and B (finite or infinite) have same cardinality ifthere is a bijective function f : A→ B.

I Two sets having same cardinality are also called asnumerically equivalent sets.






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I A set A is called denumerable (or countably infinite) if|A| = |N|.

I A set which is finite and denumerable is said to be countable.

I A set that is not countable is called uncountable.

I Theorem: A countable union of countable sets is countable.

I For any sets A and B, we define |A| ≤ |B| if there exist aone-to-one function f : A→ B.

I |A| < |B| if |A| ≤ |B| but |A| 6= |B|.I Theorem: (Cantor) For any set A, we have|A| < |power(A)|.

I Theorem: If |A| ≤ |B| and |B| ≤ |A| then |A| = |B|.






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I |A| < |B| if |A| ≤ |B| but |A| 6= |B|.

I Theorem: (Cantor) For any set A, we have|A| < |power(A)|.







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Functions and Algorithms - McMaster Universityms.mcmaster.ca/~nahid/Math-301/Lecturenotes/... · Exponential and logarithmic functions Sequence and Summation Sequence Summation Indexed