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**Cardinality and Algebraic Structures**

Dr Tijl De Bie Dept. Eng. Maths.

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**Contents Part I (weeks 1-7) 1 Introduction**

2 Combinatorics, permutations and combinations. 3 Algebraic Structures and matrices: Homomorphism, isomorphism, group, semigroup, monoid, rings, fields 4 Lattices and Boolean algebras If time remains: some illustrations of the use of group theory in cryptography Part II (weeks 8-12) Vector spaces

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**Introduction Computer programs frequently handle real world data.**

This data might be financial e.g. processing the accounts of a company. It may be engineering data e.g. from sensors or actuators in a robotic system. It may be scientific data e.g. weather data or geological data concerning rock strata. In all these cases data typically consists of a set of discrete elements. Furthermore there may exist orderings or relationships among elements or objects. It may be meaningful to combine objects in some way using operators. We hope to clarify our concepts of orderings and relationships among elements or objects We look at the idea of formal structures such as groups, rings and and formal systems such as lattices and Boolean algebras

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Number Systems The set of natural numbers is the infinite set of the positive integers. It is denoted N and can have different representations: {1,2,3,4, } {1,10,11,100,101,.....} are alternative representations of the same set expressed in different bases. Nm is the set of the first m positive numbers i.e. {1,2,3,4, ,m}. N0 is the set of natural numbers including 0 i.e. {0,1,2,3,5,....} Q denotes the set of rational numbers i.e. signed integers and fractions {0,1,-1,2,-2,3,-3,....,1/2,-1/2,3/2,-3/2,5/2, -5/2,....,1/3,-1/3,2/3,-2/3, } R is the set of real numbers i.e. the coordinates of all the points on a line. Z is the set of all integers, both positive and negative {0,1,-1,2,-2,3,-3,......}

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**2 Combinatorics: Permutations**

A permutation of the elements of a set A is a bijection from A onto itself. If A is finite we can calculate the number of different permutations. Suppose A={a1,...,an} n choices n-1 choices 1 choice a1 a2 an total number of ways of filling the n boxes n x (n-1)x(n-2)x(n-3) x1=n! nPn=n! eg a possible permutation of {1,2,3,4,5,6} is

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**Composition of Permutations**

If :A A and :A A are permutations of A then the composition or product .of and satisfies for all x in A .x)= (x)) Notice that since both and are bijections from A into A so is . In other words . is a permutation of A. Example: Let A={1,2,3,4,5,6} then two possible permutations are For . we have that

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**Cyclic Permutations A cyclic permutation on a set A of n elements has**

the form where : For shorthand we often write is said to be a k cycle Example or (6 1 4) is a cyclic permutation Two cyclic permutations and are said to be disjoint if e.g. (4 5 2) and (3 1 6) are disjoint

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Notice that Other examples are or Can you spot a product of disjoint cyclic permutations equivalent to the following permutation ?

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**Theorem: Every permutation of a finite set A can be expressed as a combination of disjoint cycles.**

Structure underlying permutations Note that the following hold: (1) The product of two permutations is a uniquely determined permutation of the same set. (2) The composition of permutations is associative. (3) The permutation is called the identity permutation and has the property that (4) For every permutation there is an inverse such that

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Combinations When we think about combinations we do not allow repeats and unlike permutations we do not consider order. Combinations look at the number of different ways of picking a subset of k elements from a set of n elements. Think of the number of ways of picking a list of k distinct elements of n no. of choices n n-1 n-k-2 n-k-1 places = n(n-1)(n-2) (n-k-1) = n!/(n-k)! For each possible list there are k! permutations so since we are not interested in order we should divide the above by k!. C(n,k) = Cnk = n!/(n-k)!k!

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**Example: Choosing 2 elements from {a,b,c,d}**

{a,b},{a,c},{a,d}, {b,c},{b d},{c,d} C(4,2)= 4!/(2! 2!) =6 Combinations with Repetitions We could also consider combinations with repetitions. With repetitions the number of distinct combinations of k elements chosen from n is: C(n+k-1,k)= (n+k-1)!/k!(n-1)! Number of different throws of 2 identical dice (1 1)(2 2)(3 3)(4 4)(5 5)(6 6) (1 2)(1 3)(1 4)(1 5)(1 6) (2 3)(2 4)(2 5)(2 6) (3 4)(3 5)(3 6)(4 5)(4 6)(5 6) C(7,2)=21

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Algebraic Structures When we consider the behaviour of permutations under the composition operation we noticed certain underlying structures. Permutations are closed under this operation, they exhibit associativity, an identity element exists and an inverse exists for each permutation These properties define a general type of algebraic structure called a group. In this section we shall look at groups in more detail as well as other similar algebraic structures such as semigroups and monoids. Later we will progress to consider more complex algebraic structures such as rings, integral domains and fields. We will see that many real life situations are examples of these algebraic structures

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**Groups A group or is a set G with binary operation**

which satisfies the following properties 1. is a closed operation i.e. if and then 2. this is the associative law 3. G has an element e, called the identity, such that 4. there corresponds an element such that Example: The set of all permutations of a set A onto itself is group (called the symmetric group Sn for n elements).

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**Group of Symmetries of a Triangle**

Consider the triangle X O Y Z n m We can perform the following transformations on the triangle 1=identity mapping from the plane to itself p=rotation anticlockwise about O through 120 degrees q=rotation clockwise about O through 120 degrees a=reflection in l b=reflection in m c=reflection in n

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Let denote transformation y followed by transformation x for x and y in {1,p,q,a,b,c} So for example l l l Y X X a p O O O X Z m n m Y Z n m Z Y n Notice the table is not symmetric

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**Other examples of a group**

The set of all permutations onto itself is a group (called the symmetric group Sn) The sets of all invertible nxn matrices forms a group under ordinary matrix multiplication (called GL(n), the general linear group) The quaternion group: Let G={I,-I,J,-J,K,-K,L,-L} where I=[ ], J=[ ], K=[ ] , L=[ ] j j 0 j j 0

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**Order of a group A finite group is a group where G is finite**

The order of a finite group is |G| For example if G is the set of permutations of a set A with n elements then the order of G is n!

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**Abelian Groups If is a group and is also commutative then**

is referred to as an Abelian group (the name is taken from the 19’th century mathematician N.H. Abel) is commutative means that Examples: and are abelian groups. Why is not a group at all?

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**Modular arithmetic Recall a=b mod p iff p|a-b**

Notice a=b mod p iff a=kp+b for some integer k a=b mod p implies p|a-b implies a-b=kp implies a=kp+b a=kp+b implies a-b=kp implies p|a-b implies a=b mod p

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Modular addition Modular addition mod 6: + 1 2 3 4 5

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**Modular multiplication**

Modular multiplication mod 7: x 1 2 3 4 5 6

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**Modular multiplication**

Modular multiplication mod 6: x 1 2 3 4 5

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**Modular multiplication**

Not a group! (Why not?) Which subset of {1,2,3,4,5} does form a group? x 1 2 3 4 5

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**Modular multiplication**

Theorem: If n>=2 and n|p then n has no inverse under multiplication mod p Prove it! The subset of {1,…,p-1} relatively prime to p is a group under multiplication mod p denoted Zp* We will clarify this on the next slides…

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Modular arithmetic Recall Euclid’s algorithm to find the gcd of x and y: x=k1y+r1 y=k2r1+r2 r1=k3r2+r3 … rn-2 =kn-1rn-1+rn rn-1=knrn From this… Theorem: There exist integer a and b such ax+by=gcd(x,y) The old remainder is divided by the new one repeatedly until the remainder is 0 The gcd is the last non zero remainder

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Modular arithmetic An element n has an inverse n-1 under multiplication mod p for which n. n-1 =1 mod p if and only if (iff) n is relatively prime to p. Prove this! Clearly then if p is prime then every element will have an inverse.

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**Groups in logic Consider exclusive or defined by**

A⊕ B ≡(¬A∧ B)∨ (A∧ ¬B) {t,f} is an abelian group under exclusive or. What is the identity? What is the inverse of t (and f)?

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**To show that an algebraic system is a group we**

must show that it satisfies all the axioms of a group. Question: Let be a Boolean algebra so that A is a set of propositional elements, is like ‘or’, is like ‘and’ and is like ‘not’. Show that is an abelian group where Answer: (1) Associative since prove this ? (2) Has an identity element 0 (false) since (3) Each element is its own inverse (4) The operation commutes prove this ?

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**Iterated operations a=a1 a◦a=a1 a◦a◦a=a2 a◦a…◦a=ak**

(Why is this unambiguously defined?)

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Cyclic groups A group G is cyclic if there exists a∈G such that for any b∈G there is an integer k≥0 such that ak=b. I.e. Every element of G is some power of a. Element a is called the generator of G denoted G=<a> Example: <{1,-1},×>=<-1> since –12=1, -13=-1

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**Order of a cyclic permutation group**

Show that the order is equal to p [Show by making a drawing…]

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Weaker structures An Abelian group is a strengthening of the notion of group (i.e. requires more axioms to be satisfied) We might also look at those algebraic structures corresponding to a weakening of the group axioms Semigroup ⊆ monoid ⊆ group ⊆ Abelian Group

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**Semigroup is a semigroup if the following conditions are satisfied: 1.**

is a closed operation i.e. if and then 2. is associative Example: The set of positive even integers {2,4,6,.....} under the operation of ordinary addition since The sum of two even numbers is an even number + is associative The reals or integers are not semigroups under - why?

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**Monoid is a monoid if the following conditions are satisfied: 1.**

is a closed operation i.e. if and then 2. is associative 3. There is an identity element Examples: Let A be a finite set of heights. Let be a binary operation such that is equal to the taller of a and b. Then is a monoid where the identity is the shortest person in A is a monoid: is associative, true is the identity, but false has no inverse is a monoid: is associative false is the identity, but true has no inverse

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**Properties of Algebraic Structures**

Theorem: (unique identity) Suppose that is a monoid then the identity element is unique Proof: Suppose there exist two identity elements e and f. [We shall prove that e=f] Theorem: (unique inverse) Suppose that is a monoid and the element x in A has an inverse. Then this inverse is unique. Proof: ??

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**Properties of Groups Theorem (The cancellation laws): Let be**

a group then (i) (ii) Proof: (i) Suppose that then by axiom 3 a has an identity and we have that (ii) is proved similarly Theorem (The division laws): Let be a group then (i) (ii) Proof ??

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**Theorem (double inverse) :If x is an element of**

the group then Proof: Theorem (reversal rule) If x and y are elements of the group then Proof ??

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**For an arbitrary element of a group**

we can define functions and such that Theorem: and are permutations of G Proof: Consider [prove 1-1] suppose for x,y in G [Prove onto] For any y in G Corollary: In every row or column of the multiplication table of G each element of G appears exactly once.

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**Subgroups is a subgroup of the group if and is also a group Examples:**

Test for a subgroup Let H be a subset of G. Then is a subgroup of iff the following conditions all hold: (1) (2) H is closed under multiplication (3) For every group , and are subgroups is called the trivial subgroup of a proper subgroup of is a subgroup different from G A non-trivial proper subgroup is a subgroup equal neither to or to

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**Cosets Consider a set A with a subset H. Let .**

Then the left coset of H with respect to a is the set of elements: This is denoted by Similarly the right coset of H with respect to a is and is denoted by Example: Let A be the set of rotations and {0º,120º,240º} . Let then which is the right coset with respect to

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**Normal Subgroups Let be a subgroup of . Then**

is a normal subgroup if, for any , the left coset is equal to the right coset is a normal subgroup where e.g. Theorem: In an Abelian group, every subgroup is a normal subgroup

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Coset cardinality Theorem: For any H subset of G and any a in G |a•H|=|H| Proof: By definition of Coset |a•H|≤|H| Now suppose |a•H|<|H| then there must exist b and c distinct elements of H such that a•b=a•c. But by the cancellation law this implies that b=c which is a contradiction. Hence |a•H|=|H|

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Coset partitioning Theorem: Let a,b∈G and let H be a subgroup of G then either: a•H=b•H or: a•H∩ b•H=∅ Proof: Suppose a•H∩ b•H≠∅ then there exist s and t in H such that a•s=b•t. In this case a= b•t•s-1 and for an arbitrary x in H a•x= b•t•s-1•x Now by the inverse axiom and closure, t•s-1•x∈H and hence b•t•s-1•x∈b•H, therefore a•x∈b•H so that a•H⊆b•H Similarly we can show that b•H⊆a•H Hence if the two cosets are not disjoint then b•H=a•H

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LeGrange’s theorem Theorem: Let H be a subgroup of finite group G, then the cardinality of H evenly divides the cardinality of G (i.e |H| | |G|) Proof Let |G|. Now for each element ai of G we can generate a coset ai•H. Now notice that ai∈ai•H because since H is a subgroup, e∈H and ai•e= ai Suppose there are m distinct cosets of H then picking one representative ai from each this means that: G= a1•H∪ a2•H ∪ a3•H … ∪ am•H

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LeGrange’s theorem Now by the previous theorem it follows that since these m cosets are distinct then they must be disjoint. Hence, |G|=|a1•H|+ |a2•H| + |a3•H| … + |am•H| Also by the cardinality theorem for cosets they all have the same cardinality, namely |H|. Hence, |G|=m.|H| as required

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Order of an element Let i be the smallest integer such that ai=e where a is an element of group G and e is the identity element. If i exists we call it the order of a. Otherwise we say that a has infinite order.

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**Subgroup generated by an element**

Theorem: For any element a of G with finite order the set: H={aj: for some integer j} is a subgroup of G. Notice: if i is the order of element a then ai=e ai+1=e•a=a1 ai+2= a •a =a2 ai+n=an

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**Example Let σ=(1 2 3 4), a permutation of 4 elements**

Then {σ, σ2, σ3, σ4} is a subgroup of the group of permutations of {1,2,3,4} The order of σ is 4 [Work it out!]

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**Order of elements in finite groups**

If the group G is finite then all elements of G have finite order: For any a∈G, since G is finite there must exist i<j such that ai=aj a•ai-1=a•aj-1 cancellation law implies ai-1=aj-1 Repeated application of the cancellation law gives a=aj-i+1 a•e=a•aj-i implies e=aj-i

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Corollary of LeGrange Theorem: The order of every element of a finite group G, divides the order of G Proof... Every element of G has finite order n and hence generates a subgroup of order n. Hence by LeGrange’s theorem n divides |G|

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Isomorphism Two groups are isomorphic if there is a bijection of one onto the other which preserves the group operations i.e. if and are groups then a bijection is an isomorphism provided Example: Consider the group of matrices of the form where under matrix multiplication. This is isomorphic to the group The mapping is An isomorphism from a group onto itself is called an automorphism.

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Homomorphisms The idea of isomorphic algebraic structures can be readily generalised by dropping the requirement that the functional mapping be a bijection. Let and be two algebraic systems then a homomorphism from to is a functional mapping such that Example: consider the two structures then f such that is a homomorphism between and

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**Algebraic Structures with two Operations**

So far we have studied algebraic systems with one binary operation. We now consider systems with two binary operations. In such a system a natural way in which two operations can be related is through the property of distributivity; Let be an algebraic system with two binary operations and . Then the operation is said to distribute over the operation if and Example: distributes over + distributes over distributes over

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**Ring An algebraic system is called a ring if**

the following conditions are satisfied: (1) is an Abelian group (2) is a semigroup (3) The operation is distributive over the operation Example: is a ring since is an Abelian group is a semigroup distributes over +

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**Examples of rings <Z, +, ×> is a ring because:**

<Z, +> is an Abelian group. <Z, ×> is a semigroup. × distributes over + The set {[ ],a,b є R} is a ring under matrix addition and multiplication {0,1,…,n-1} is a ring under addition and multiplication mod n 0 a 0 b

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Rings of polynomials Let the set R[x] be the set of all polynomial of the form: anxn+…+ a2x2+ a1x1+a0 for some n, where an,…,a0 єR Then R[x] is a ring under addition and multiplication of polynomials In fact for any ring R you can construct a ring of polynomials R[x] over R

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**Special types of ring A commutative ring is a ring in which is**

A ring with unity contains an element 1 such that (0 is the identity of ) Example: the ring of 2x2 matrices under matrix addition and multiplication is a ring with unity. The element 1=I=

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Division rings A division ring is a (not necessarily commutative) ring with unity, in which every element a not equal to 0 has an inverse a-1 such that a•a-1= a-1•a=1 The ring of complex matrices of the form: [ ] a b -b a

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**Integral Domains and Fields**

is an integral domain if it is a commutative ring with unity that also satisfies the following property; is also an integral domain is a field if: (1) is an Abelian group (2) is an Abelian group (3) The operation is distributive over the operation Example:The set of real numbers with respect to + and is a field. is not a field. Why?

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Galois fields For a prime number p the set {0,1,…,p-1} is a field under modular addition and multiplication mod p A field (like this one) with finite number of elements is called a Galois field.

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**A Field is an Integral Domain**

Let be a field then certainly is a commutative ring with unity. Hence, it only remains to prove that Now suppose then if x=0 the above holds. Consider the case then where Since is an Abelian group then it must contain an inverse to x, , for which the following holds Now Therefore y=0 as required

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**Properties of a ring Theorem: if is a ring. Then**

Proof: as for previous argument Let -x denote the inverse of x under Theorem: if is a ring then the following hold (i) (ii) Proof: (i)

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(ii) for both (i) and (ii) the symmetric cases are proved similarly

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**Property of an integral domain**

Theorem: suppose that elements a,b and c of an integral domain satisfy and then b=c. Proof:

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**Subrings and subfield Subring**

If (A,⊕,•) is a ring then (H,⊕,•) is a subring if H⊆A and (H,⊕,•) is a ring Subfield If (A,⊕,•) is a field then (H,⊕,•) is a subfield if H⊆A and (H,⊕,•) is a field Examples: Z is a subring of R, R is a subfield of C

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Ring morphisms A morphism between rings (A,⊕,•) and (B,*,⊗) is a function f:A→B such that: ∀x,y∈A f(x⊕y)=f(x)*f(y) and f (x•y)=f(x)•f(y) From these we have that f(0)=0′ where 0′ is the zero of (B,*,⊗) Also f(-x)=-f(x)

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**Special morphisms An injective ring morphism is called a monomorphism**

2. A surjective ring morphism is called an epimorphism 3. A bijective ring morphism is called a isomorphism

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Examples of morphisms f(a) = a mod n, is an epimorphism (surjective ring morphism) between Z and {0,1,…,n-1} For the ring of polynomials R(x), f(p)=p(j) is an epimorphism into C, where p(j) is obtained by substituting j for x in the polynomial p

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Galois theorem For every prime power pk (k=1,2,…) there is a unique (upto isomorphism) finite field containing pk elements denoted by GF(pk) All finite fields have cardinality pk

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**Galois theorem: examples**

GF(2) + | · | | | | | 0 1 GF(3) + | · | | | | | | | 0 2 1

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**Partial Orderings We have introduced formal structure governing**

the properties of various sets of elements under one or two binary operations. These elements can also be ordered and restricted by binary relations. In this section we revise our understanding of binary relations in a set and also introduce a graphical notation for binary relations. A relation R on a set A is a partial order if it satisfies; (1) R is reflexive (2) R is antisymmetric (3) R is transitive The pair (A,R) is called a partially ordered set or poset Example: Set of reals R with the relation

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Example: The relation can be defined on a Boolean algebra by; (1) Thus from the idempotent law we find that and hence the relation is reflexive. (2) If From the commutative law and hence the relation is antisymmetric (3) If then

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**We can think of a relation as being represented by**

the set of pairs of elements which satisfy the relation. In this case a partial ordering on A corresponds to a subset B of AxA satisfying Other examples of partial orderings: Divisibility on N: We say that a divides b iff there is some x in Z such that ax=b. If this divisibility exists we write a|b. Divisibility is a partial order on N. Inclusion on a set of sets X

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**Graphical Representations**

We can represent partial orderings graphically by means of a directed graph where the nodes are elements of A and the directed edges give the partial order relations. e.g. the graph a b c d Denotes the partial ordering on {a,b,c,d} where

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**Graphical Representations of the Axioms**

Reflexive: a Antisymmetric: the following does not occur a b Transitive: a b c

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**Example: Divisibility relation on{2, 3, 4, 6, 8, 9, 18}**

2|4 4|8 2|8 2|6 3|6 3|9 9|18 6|18 3|18 2|18 2 4 8 6 3 9 18

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**Example: The collection of all subsets of {a,b,c}**

{a,c} {b,c} {a} {b} {c}

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**Hasse Diagrams Notice that some of the diagrams in the previous**

examples were messy and difficult to read having many links. We can simplify these diagrams by introducing certain conventions. The Hasse diagram of a partially order set is a drawing of the points in the set (as nodes) and some of the links of the graph of the order relation. The rules for drawing the Hasse diagram of a partial order are: (1) Omit all links that can be inferred from transitivity. (2) Omit all loops (3) Draw links without arrow heads (4) Understand that all arrows would point upwards

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**Here are Hasse diagrams for the two examples**

given previously: Divisibility: 18 8 9 4 6 3 2

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Example: subsets {a,b,c} {a,b} {a,c} {b,c} {a} {b} {c}

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**Incomparable Elements**

Consider the Hasse diagram for divisibility on {2,3,....,10} 10 8 6 9 4 5 3 7 2 Notice that 5 and 6 are not related in either direction Similarly for 2 and 3 If neither R(a,b) or R(b,a) then a and b are incomparable or not comparable

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Linear or Total Order A linear or total order on a set A is a partial order on A in which every two elements are comparable 5 4 3 2

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**Maximal and Minimal Elements**

A maximal element of A is any element t of A such that A minimal element of A is any element b of A such that Example:For the subset ordering {a,b,c} is the maximal element and is the minimal element For divisibility on {2,.....,10} the maximal elements are 6, 7, 8, 9 and 10 and the minimal elements are 2, 3, 5 and 7 The element 4 is neither maximal nor minimal

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**Upper Bounds and Lower Bounds**

Let S be a subset of A then x in A is an upper bound of S if Similarly z in A is a lower bound of S if An element u is the least upper bound of S if u is an upper bound of S and for every x an upper bound of S R(u,x) An element l is the greatest lower bound of S if l is an upper bound of S and for every z a lower bound of S R(z,l) The least upper bound (lub) of S is sometimes referred to as the supremum of S (sup S) The greatest lower bound (glb) of F is sometimes referred to as the infimum of S (inf S)

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**Lattices A partially ordered set in which every pair of**

elements has a least upper bound and a greatest lower bound is called a lattice. a b c d f e This is not a lattice since {c,d} has no lub or glb. A lattice in which every subset has a lub and glb is called complete. Every finite lattice is complete. For a complete lattice the lub of the whole lattice is call top and the greatest lower bound bottom

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**Example: Consider elements of the form (a,b,c)**

where a,b and c can take the values 0 or 1. For two such elements f and g we say that if each coefficient of f is less than or equal to the corresponding coefficient of g e.g. but not (111) (101) (011) (110) (001) (100) (010) (000)

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**Meet and Join In a lattice the following equations**

define binary operations on A is called the meet operation and is called the join operation. They have the following properties Commutativity: Associativity: Since is an upper bound of a and b Similarly for the meet

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Theorem and If then Proof Let 1 denote the lub of the whole lattice and 0 denote the glb of the whole lattice. Then

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**Example Let us order the following set of numbers with the**

operation “is a factor of”. A={3,9,12,15,36,45} 45 36 9 15 12 3 The join operation is the least common multiple The meet operation is the greatest common divisor

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**Complemented Lattice For a complemented lattice we have that for**

there exists such that: e.g. 1 a b c

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**Distributive Lattice A lattice is distributive if:**

e.g. the following lattice is not distributive e d b c a Since

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**Boolean Algebra A Boolean Algebra consists of two binary**

operations and and the unary operation on a set B with distinct elements 0 and 1 such that the following hold. (1) The commutative laws: (2) The associative laws: (3) The Distributive laws: (4) The Identity Laws:

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**(5) The Complementation Laws:**

Theorem If is a complemented distributive lattice then is a Boolean algebra where correspond to the meet, join and complement operations on L respectively Proof ?

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**(6) The following Idempotent Laws can be derived:**

Proof (7) The following Identity Laws can also be derived Proof

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