A short presentation to explain the use of permutations and combinations and some examples to illustrate the concepts. This was made as an assignment in which i was to explain the concepts to the class.
SET
A set is a well defined collection of objects, called the “elements” or “members” of the set.
A specific set can be defined in two ways-
If there are only a few elements, they can be listed individually, by writing them between curly braces ‘{ }’ and placing commas in between. E.g.- {1, 2, 3, 4, 5}
The second way of writing set is to use a property that defines elements of the set.
e.g.- {x | x is odd and 0 < x < 100}
If x is an element o set A, it can be written as ‘x A’
If x is not an element of A, it can be written as ‘x A’
Special types of sets-
Standard notations used to define some sets:
N- set of all natural numbers
Z- set of all integers
Q- set of all rational numbers
R- set of all real numbers
C- set of all complex numbers
TYPES OF SETS
-subset
-singleton set
-universal set
-empty set
-finite set
-infinte set
-eual set
-disjoint set
-cardinal set
-power set
OPERATIONS ON SET
The four basic operations are:
1. Union of Sets
2. Intersection of sets
3. Complement of the Set
4. Cartesian Product of sets
Union of two given sets is the smallest set which contains all the elements of both the sets.
A B = {x | x A or x B}
Let a and b are sets, the intersection of two sets A and B, denoted by A B is the set consisting of elements which are in A as well as in B
A B = {X | x A and x B}
If A B= , the sets are said to be disjoint.
If U is a universal set containing set A, then U-A is called complement of a set.
THE BINOMIAL THEOREM shows how to calculate a power of a binomial –
(x+ y)n -- without actually multiplying out.
For example, if we actually multiplied out the 4th power of (x + y) --
(x + y)4 = (x + y) (x + y) (x + y) (x + y)
-- then on collecting like terms we would find:
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 . . . . . (1)
This powerpoint was used in my 7th and 8th grade classes to review the fundamental counting principle used in our probability unit. There are three independent practice problems at the end.
A short presentation to explain the use of permutations and combinations and some examples to illustrate the concepts. This was made as an assignment in which i was to explain the concepts to the class.
SET
A set is a well defined collection of objects, called the “elements” or “members” of the set.
A specific set can be defined in two ways-
If there are only a few elements, they can be listed individually, by writing them between curly braces ‘{ }’ and placing commas in between. E.g.- {1, 2, 3, 4, 5}
The second way of writing set is to use a property that defines elements of the set.
e.g.- {x | x is odd and 0 < x < 100}
If x is an element o set A, it can be written as ‘x A’
If x is not an element of A, it can be written as ‘x A’
Special types of sets-
Standard notations used to define some sets:
N- set of all natural numbers
Z- set of all integers
Q- set of all rational numbers
R- set of all real numbers
C- set of all complex numbers
TYPES OF SETS
-subset
-singleton set
-universal set
-empty set
-finite set
-infinte set
-eual set
-disjoint set
-cardinal set
-power set
OPERATIONS ON SET
The four basic operations are:
1. Union of Sets
2. Intersection of sets
3. Complement of the Set
4. Cartesian Product of sets
Union of two given sets is the smallest set which contains all the elements of both the sets.
A B = {x | x A or x B}
Let a and b are sets, the intersection of two sets A and B, denoted by A B is the set consisting of elements which are in A as well as in B
A B = {X | x A and x B}
If A B= , the sets are said to be disjoint.
If U is a universal set containing set A, then U-A is called complement of a set.
THE BINOMIAL THEOREM shows how to calculate a power of a binomial –
(x+ y)n -- without actually multiplying out.
For example, if we actually multiplied out the 4th power of (x + y) --
(x + y)4 = (x + y) (x + y) (x + y) (x + y)
-- then on collecting like terms we would find:
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 . . . . . (1)
This powerpoint was used in my 7th and 8th grade classes to review the fundamental counting principle used in our probability unit. There are three independent practice problems at the end.
Euclid's Algorithm for Greatest Common Divisor - Time Complexity AnalysisAmrinder Arora
Euclid's algorithm for finding greatest common divisor is an elegant algorithm that can be written iteratively as well as recursively. The time complexity of this algorithm is O(log^2 n) where n is the larger of the two inputs.
This Slideshare presentation tells you how to tackle with questions based on number of theory. Ample of PPT of this type on every topic of CAT 2009 are available on www.tcyonline.com
The simplest and most common form of mathematical induction infers that a statement involving a natural number n (that is, an integer n ≥ 0 or 1) holds for all values of n. The proof consists of two steps:
The base case (or initial case): prove that the statement holds for 0, or 1.
The induction step (or inductive step, or step case): prove that for every n, if the statement holds for n, then it holds for n + 1. In other words, assume that the statement holds for some arbitrary natural number n, and prove that the statement holds for n + 1
This PPT discusses about some programming puzzles that are related to Encryption and also it emphasis the need for strengthening bit-wise operators concept.
In this note we, first, recall that the sets of all representatives of some special ordinary residue classes become (m, n)-rings. Second, we introduce a possible p-adic analog of the residue class modulo a p-adic integer. Then, we find the relations which determine, when the representatives form a (m, n)-ring. At the very short spacetime scales such rings could lead to new symmetries of modern particle models.
Solutions Manual for An Introduction To Abstract Algebra With Notes To The Fu...Aladdinew
Full download : https://goo.gl/XTCXti Solutions Manual for An Introduction To Abstract Algebra With Notes To The Future Teacher 1st Edition by Nicodemi
"Hiring Great Technologists in Six Easy Steps"
In this talk, we discuss:
- What it takes to hire and retain great engineers in New York area.
- Qualities you should look for in a technical co-founder.
- Does it make sense to outsource?
- As well as other topics raised by the audience.
Afterwards, we will continue with networking among the group's members.
Please feel free to write your questions ahead of time in the comments section.
You Too Can Be a Radio Host Or How We Scaled a .NET Startup And Had Fun Doing ItAleksandr Yampolskiy
Cinchcast (aka BlogTalkRadio) is a startup in New York City.
Using only a phone, you can broadcast your message globally to millions of listeners.
Thousands of broadcasts are happening every day on topics ranging from technology to battling cancer.
In this talk, we will discuss how we accomplished this, the technology behind it, and the challenges ahead.
We will talk about what it's like building a startup in .NET and the techniques we have used to scale, such as
HTML and donut caching, lazy loading of data, elastic search, as well as marrying telephony to the web stack.
This talk describes the benefits of social media as well as its security challenges. It also outlines sample defenses that companies can adopt. It was given at CSO breakfast club in NYC.
3. Divisors
n A non-zero number b divides a if 9m
s.t. a=mb (a,b,m 2 Z)
n That is, b divides into a with no
remainder
n We denote this b|a
n Example:
¡ all of 1,2,3,4,6,8,12,24 divide 24
¡ 6 | 24 (4*6 = 24), 1 | 24 (24*1 = 24),
but 5 | 24 (no m2Z such that m*5 = 24)
4. Divisors (cont.)
n Some axioms:
¡ a|1 ) a = §1
¡ a|b Æ b|a ) a = §b
¡ 8b?0 b|0
¡ b|g Æ b|h ) b|(mg + nh)
n A number p is prime , p ? 1 Æ 8m2
(1, p) m | p
5. Groups
n Def: A set G with a binary operation
?: G£ G ! G is called a group if:
1. (closure) ∀ a,b∈G, a?b∈G
2. (associativity) ∀ a,b,c∈G, (a?b)?c=a?(b?c)
4. (identity element) ∃ e∈G, ∀ a∈G, a?e=a
5. (inverse element) ∀ a∈G, ∃ a-1∈G, a?(a-1)=e
n A group is commutative (Abelian) if
∀ a,b∈G, a?b=b?a
6. Examples of groups
n Integers under addition, (Z, +) = {…, -2, -1, 0, +1,
+2, …}.
Identity: e = 0. Inverses: a-1 = -a
n ({Britney, Dustin}, ? ), where
¡ Britney? Britney = Britney
¡ Britney? Dustin = Dustin
¡ Dustin ? Britney = Dustin
¡ Dustin ? Dustin = Britney
Identity: e = Britney. Inverses: Britney -1 = Britney, Dustin-1 =
Dustin.
7. Subgroups
n Let (G, ?) be a group. (H, ?) is a sub-
group of (G, ?) if it is a group, and
H⊆G.
n Lagrange’s theorem: if G is finite and
(H, ?) is a sub-group of (G, ?) then |H|
divides |G|
8. Cyclic groups
n We define exponentiation as repeated
application of operator ?. For example,
¡ a3 = a?a?a
¡ we also define a0 = e and a-n = (a-1)n
n A group G is cyclic if every element is a
power of some fixed element.
n That is, G = <a> = {e, a, a2, a3,…} for some
a.
n a is said to be a generator of the group
9. A theorem…
Theorem: If (G, ? ) is a finite group, then
8a2 G a|G| = e.
Proof:
¡ Fix a2G. Consider <a> = {a0 = e, a, a2, …}
¡ |G| < 1 Æ <a> = G ) |<a>| < 1
¡ Hence, <a> = {e, a, a2, …, ak-1} for some k and
ak = e.
¡ By Lagrange’s Theorem, |<a>| divides |G| )
|G| = d¢|<a>| for some d2 Z.
¡ So, a|G| = ad¢|<a>| = ad¢k = {ak} d = ed. QED.
10. Rings
n Def: A set R together with two operations (+, ?) is a ring if
1. (R, +) is an Abelian group.
2. (R, ?) is a semi-group (just needs to be
associative)
3. ? distributes over +: a(b + c) = ab + ac and
(a + b)c = ac + bc
n We use +, ?, only for the sake of using familiar and intuitive
notation. We could instead use any symbols. We are
NOT doing regular addition/multiplication.
n In the ring R, we denote by: -a, the additive inverse of a.
On commutative rings, the multiplicative inverse of a is
denoted by a-1 (when it exists).
11. Rings (cont.)
n Example: set of 2x2 matrices forms a
ring under regular matrix (+, *).
n Some questions to think about:
¡ Is it always the case that A + B = B + A?
¡ What about A*B = B*A?
¡ What is the identity element?
12. Fields
n Def: A field is a commutative ring with
identity where each non-zero element has a
multiplicative inverse: ∀ a≠0∈F, ∃ a-1∈F,
a·a-1=1
n Equivalently, (F,+) is a commutative
(additive) group and (F {0}, ·) is a
commutative (multiplicative) group.
n Example: set of rational numbers Q
13. Modular arithmetic
n Def: Modulo operator a mod n = remainder
when a is divided by n
(Another notation: a % n)
n Example: 11 mod 7 = 4, 10 mod 5 = 0, 3 mod 2
= 1. n-1 0 1
.
.
.
01 n-1 n
clock
arithmetic
14. Modular arithmetic (cont.)
n a is congruent to b (a = b mod n) if
when divided by n, a and b give the
same remainder (a mod n = b mod n)
n a ´ b mod n if n | (a – b)
n E.g. 100 ´ 34 mod 11
15. Zn
n a´ b mod n defines an equivalence
relation
n set of residues Zn = {0, 1, …, n-1}
n Each integer r2 Zn actually represents
a residue class [r] = {a2 Z : a ´ r mod
n}
17. Zn (cont.)
n Integers mod n Zn = {0, 1, …, n-1} is
an Abelian group.
n Example: What is 3+5 in Z7? What is
-6 in Z7?
n Note some peculiarities for Zn
¡ if (a+b)=(a+c) mod n then b=c mod n
¡ but (ab)=(ac) mod n then b=c mod n only
if a is relatively prime to n
18. Zn*
n Multiplicative integers mod n
Zn* = {x2 Zn : gcd(x, n) = 1}
n Zn* consists of all integers 0…n-1
relatively prime with n
n What is the size of this group? Euler’s
totient function φ(n) = |Zn*|
19. Zn* (cont.)
n What is φ(p) when p is prime?
¡ ZP* = {1, 2, …, p-1} ) φ(p) = |Zp*| = p – 1.
n What about φ(pk) where p is prime and k >
1?
¡ Zpk = {0, 1, …, pk – 1}
¡ How many multiples of p are in Zpk?
¡ Multiples are {0, p, 2p, …, (pk-1 – 1)p}. There are
pk-1 of them
¡ Hence, φ(pk) = pk – pk-1
20. Zn* (cont.)
n φ(mn) = φ(m)¢ φ(n)
n φ(∏i pie) = ∏i(pie – pie-1)
n Example:
¡ φ(10) = φ(2)¢φ(5) = 1¢4 = 4
¡ S = {1· n · 10 : n relatively prime to 10} =
{1. 3, 7, 9}. Notice that |S| = 4 as expected.